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B-series-sympy.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this notebook we use `bseries.jl` to investigate error expansions for RK methods applied to specific problems."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# Load the packages we will use. These must first be installed using: import Pkg; Pkg.add(\"package_name\")\n",
"using BSeries\n",
"using Latexify\n",
"using RootedTrees\n",
"using Symbolics\n",
"import SymPy; sp=SymPy;"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"First we specify the Butcher coefficients of the RK method. This can include symbolic expressions and parameterized families of methods."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{h^{2}}{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{h^{3}}{8 \\alpha} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{}$"
],
"text/plain": [
"L\"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{h^{2}}{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{h^{3}}{8 \\alpha} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{}$\""
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Ξ± = sp.symbols(\"Ξ±\", real=true)\n",
"A = [0 0; 1/(2*Ξ±) 0]; b = [1-Ξ±, Ξ±]; c = [0, 1/(2*Ξ±)]\n",
"coeffs = bseries(A,b,c,3)\n",
"latexify(coeffs, cdot=false)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{1}{2} h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + \\frac{1}{8} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + \\frac{1}{16} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + \\frac{-21246894637}{49670350848} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + \\frac{-722476128287}{1390769823744} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + \\frac{1970748171909370823}{42730909364772720000} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + \\frac{3898363669}{40242182400} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$"
],
"text/plain": [
"L\"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + \\frac{1}{2} h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + \\frac{1}{6} h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + \\frac{1}{8} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + \\frac{1}{24} h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + \\frac{1}{16} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + \\frac{-21246894637}{49670350848} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + \\frac{-722476128287}{1390769823744} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + \\frac{1970748171909370823}{42730909364772720000} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + \\frac{3898363669}{40242182400} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + \\frac{1}{48} h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$\""
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = Rational{Int128}[0 0 0 0 0 0 0 0;(-1//6) (1//2) 0 0 0 0 0 0;(-1//10) (1//10) (1//2) 0 0 0 0 0;(-21463//39375) (21017//26250) (-5//9) (1//2) 0 0 0 0;(-59588//54675) (118717//36450) (-4375//2187) 0 (1//2) 0 0 0;(-19993033//9443328) (28508695//3147776) (-13577105//2360832) (-4090625//3147776) (1136025//3147776) (1//2) 0 0;(367020141781//199294617600) (814214904871//22143846400) (-29834937659//1992946176) (-1983358776875//87689631744) (-6702625935//885753856) (688576//109395) (1//2) 0;(1081252805//134140608) (2639189439//74522560) (33646441//4191894) (-7873511875//210792384) (-504040617//14904512) (2110843561//115277085) (13//7) (1//2)];\n",
"b = Rational{Int128}[(1081252805//134140608),(2639189439//74522560),(33646441//4191894),(-7873511875//210792384),(-504040617//14904512),(2110843561//115277085),(13//7),(1//2)];\n",
"c = Rational{Int128}[0,(1//3),(1//2),(1//5),(2//3),(3//4),(1//4),1];\n",
"\n",
"coeffs = bseries(A,b,c,5)\n",
"latexify(coeffs, cdot=false)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Since $f$ has not been specified, the elementary differentials are indicated by the corresponding rooted tree. The rooted trees are printed as nested lists, essentially in the form used in Butcher's book. We can also print out the B-series coefficients this way:"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"TruncatedBSeries{RootedTree{Int64, Vector{Int64}}, Rational{Int128}} with 18 entries:\n",
" RootedTree{Int64}: Int64[] => 1//1\n",
" RootedTree{Int64}: [1] => 1//1\n",
" RootedTree{Int64}: [1, 2] => 1//2\n",
" RootedTree{Int64}: [1, 2, 3] => 1//6\n",
" RootedTree{Int64}: [1, 2, 2] => 1//3\n",
" RootedTree{Int64}: [1, 2, 3, 4] => 1//24\n",
" RootedTree{Int64}: [1, 2, 3, 3] => 1//12\n",
" RootedTree{Int64}: [1, 2, 3, 2] => 1//8\n",
" RootedTree{Int64}: [1, 2, 2, 2] => 1//4\n",
" RootedTree{Int64}: [1, 2, 3, 4, 5] => 1//48\n",
" RootedTree{Int64}: [1, 2, 3, 4, 4] => 1//24\n",
" RootedTree{Int64}: [1, 2, 3, 4, 3] => 1//16\n",
" RootedTree{Int64}: [1, 2, 3, 4, 2] => -21246894637//49670350848\n",
" RootedTree{Int64}: [1, 2, 3, 3, 3] => 1//8\n",
" RootedTree{Int64}: [1, 2, 3, 3, 2] => -722476128287//695384911872\n",
" RootedTree{Int64}: [1, 2, 3, 2, 3] => 1970748171909370823//213654546823863600…\n",
" RootedTree{Int64}: [1, 2, 3, 2, 2] => 3898363669//20121091200\n",
" RootedTree{Int64}: [1, 2, 2, 2, 2] => 1//2"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"coeffs"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"In this form, the rooted trees are printed as level sets. The corresponding coefficients are on the right. We can also get the B-series of the exact solution:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"TruncatedBSeries{RootedTree{Int64, Vector{Int64}}, Rational{Int128}} with 18 entries:\n",
" RootedTree{Int64}: Int64[] => 1//1\n",
" RootedTree{Int64}: [1] => 1//1\n",
" RootedTree{Int64}: [1, 2] => 1//2\n",
" RootedTree{Int64}: [1, 2, 3] => 1//6\n",
" RootedTree{Int64}: [1, 2, 2] => 1//3\n",
" RootedTree{Int64}: [1, 2, 3, 4] => 1//24\n",
" RootedTree{Int64}: [1, 2, 3, 3] => 1//12\n",
" RootedTree{Int64}: [1, 2, 3, 2] => 1//8\n",
" RootedTree{Int64}: [1, 2, 2, 2] => 1//4\n",
" RootedTree{Int64}: [1, 2, 3, 4, 5] => 1//120\n",
" RootedTree{Int64}: [1, 2, 3, 4, 4] => 1//60\n",
" RootedTree{Int64}: [1, 2, 3, 4, 3] => 1//40\n",
" RootedTree{Int64}: [1, 2, 3, 4, 2] => 1//30\n",
" RootedTree{Int64}: [1, 2, 3, 3, 3] => 1//20\n",
" RootedTree{Int64}: [1, 2, 3, 3, 2] => 1//15\n",
" RootedTree{Int64}: [1, 2, 3, 2, 3] => 1//20\n",
" RootedTree{Int64}: [1, 2, 3, 2, 2] => 1//10\n",
" RootedTree{Int64}: [1, 2, 2, 2, 2] => 1//5"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"coeffs_ex = ExactSolution(coeffs)"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + 0.5 h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + 0.125 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + 0.05 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$"
],
"text/plain": [
"L\"$F_{f}\\mathopen{}\\left( \\varnothing \\right)\\mathclose{} + h F_{f}\\mathopen{}\\left( \\rootedtree[] \\right)\\mathclose{} + 0.5 h^{2} F_{f}\\mathopen{}\\left( \\rootedtree[[]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[[]]] \\right)\\mathclose{} + 0.16666666666666666 h^{3} F_{f}\\mathopen{}\\left( \\rootedtree[[][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]]] \\right)\\mathclose{} + 0.125 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][]] \\right)\\mathclose{} + 0.041666666666666664 h^{4} F_{f}\\mathopen{}\\left( \\rootedtree[[][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[[]]]]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[][]]]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[[]]][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][][]]] \\right)\\mathclose{} + 0.03333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[][]][]] \\right)\\mathclose{} + 0.025 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][[]]] \\right)\\mathclose{} + 0.05 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[[]][][]] \\right)\\mathclose{} + 0.008333333333333333 h^{5} F_{f}\\mathopen{}\\left( \\rootedtree[[][][][]] \\right)\\mathclose{}$\""
]
},
"execution_count": 50,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"latexify(coeffs_ex,cdot=false)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Next we define our ODE. For a non-autonomous ODE, it's most convenient to just add $t$ as an additional variable. That makes the code below look a bit funny."
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"f (generic function with 1 method)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"Ξ» = SymPy.symbols(\"Ξ»\", real=true)\n",
"function f(du, u, params, t)\n",
" uu, tt = u\n",
" du[1] = Ξ»*(uu-sin(tt)) + sqrt(1-uu^2); du[2] = 1\n",
" return nothing\n",
"end"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Then we define a symbolic RHS:"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [],
"source": [
"Ξ”t = sp.symbols(\"Ξ”t\", real=true)\n",
"u, t = sp.symbols(\"u t\", real=true)\n",
"u_sym = [u, t]\n",
"f_sym = similar(u_sym); f(f_sym, u_sym, nothing, nothing)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Finally, we get the B-Series for our RK method applied to our ODE:"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\left[ \\begin{array}{r}\\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{3} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{48} + \\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{2} \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{48} - \\frac{18142497709 Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{49670350848} + \\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{48} - \\frac{722476128287 Ξ”t^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{1390769823744} + \\frac{1970748171909370823 Ξ”t^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2}}{42730909364772720000} + \\frac{3898363669 Ξ”t^{5} \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{40242182400} + \\frac{Ξ”t^{5} \\left(- Ξ» \\sin{\\left(t \\right)} + \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(- \\frac{15 u^{4}}{\\left(1 - u^{2}\\right)^{\\frac{7}{2}}} - \\frac{18 u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right)\\right)}{48} + \\frac{Ξ”t^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{24} + \\frac{Ξ”t^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{24} + \\frac{Ξ”t^{4} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{8} + \\frac{Ξ”t^{4} \\left(Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{24} + \\frac{Ξ”t^{3} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{6} + \\frac{Ξ”t^{3} \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{6} + \\frac{Ξ”t^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{2} + Ξ”t \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) + 1\\\\Ξ”t + 1\\end{array} \\right]$\n"
],
"text/plain": [
"2-element Vector{SymPy.Sym}:\n",
" Ξ”t^5*(-u/sqrt(1 - u^2) + Ξ»)^3*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/48 + Ξ”t^5*(-u/sqrt(1 - u^2) + Ξ»)^2*(Ξ»*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^2)/48 - 18142497709*Ξ”t^5*(-u/sqrt(1 - u^2) + Ξ»)*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/49670350848 + Ξ”t^5*(-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*cos(t) + (-3*u^3/(1 - u^2)^(5/2) - 3*u/(1 - u^2)^(3/2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^3)/48 - 722476128287*Ξ”t^5*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))*(Ξ»*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^2)/1390769823744 + 1970748171909370823*Ξ”t^5*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))^2/42730909364772720000 + 3898363669*Ξ”t^5*(-3*u^3/(1 - u^2)^(5/2) - 3*u/(1 - u^2)^(3/2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^2*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/40242182400 + Ξ”t^5*(-Ξ»*sin(t) + (Ξ»*(u - sin(t)) + sqrt(1 - u^2))^4*(-15*u^4/(1 - u^2)^(7/2) - 18*u^2/(1 - u^2)^(5/2) - 3/(1 - u^2)^(3/2)))/48 + Ξ”t^4*(-u/sqrt(1 - u^2) + Ξ»)^2*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/24 + Ξ”t^4*(-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^2)/24 + Ξ”t^4*(-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/8 + Ξ”t^4*(Ξ»*cos(t) + (-3*u^3/(1 - u^2)^(5/2) - 3*u/(1 - u^2)^(3/2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^3)/24 + Ξ”t^3*(-u/sqrt(1 - u^2) + Ξ»)*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/6 + Ξ”t^3*(Ξ»*sin(t) + (-u^2/(1 - u^2)^(3/2) - 1/sqrt(1 - u^2))*(Ξ»*(u - sin(t)) + sqrt(1 - u^2))^2)/6 + Ξ”t^2*(-Ξ»*cos(t) + (-u/sqrt(1 - u^2) + Ξ»)*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)))/2 + Ξ”t*(Ξ»*(u - sin(t)) + sqrt(1 - u^2)) + 1\n",
" Ξ”t + 1"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"evaluate(f_sym,u_sym,Ξ”t,coeffs)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Here's the B-Series for the exact solution of the same ODE:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\begin{equation*}\\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{3} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{120} + \\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{2} \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{120} + \\frac{7 Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{120} + \\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{120} + \\frac{Ξ”t^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{30} + \\frac{Ξ”t^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2}}{40} + \\frac{Ξ”t^{5} \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{20} + \\frac{Ξ”t^{5} \\left(- Ξ» \\sin{\\left(t \\right)} + \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(- \\frac{15 u^{4}}{\\left(1 - u^{2}\\right)^{\\frac{7}{2}}} - \\frac{18 u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right)\\right)}{120} + \\frac{Ξ”t^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{24} + \\frac{Ξ”t^{4} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{24} + \\frac{Ξ”t^{4} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{8} + \\frac{Ξ”t^{4} \\left(Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{24} + \\frac{Ξ”t^{3} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{6} + \\frac{Ξ”t^{3} \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{6} + \\frac{Ξ”t^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{2} + Ξ”t \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) + 1\\end{equation*}$\n"
],
"text/plain": [
" \n",
" 3 βŽ› βŽ› \n",
" 5 βŽ› u ⎞ ⎜ βŽ› u ⎞ ⎜ \n",
"Ξ”t β‹…βŽœ- ─────────── + λ⎟ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•±\n",
" ⎜ ________ ⎟ ⎜ ⎜ ________ ⎟ \n",
" ⎜ β•± 2 ⎟ ⎜ ⎜ β•± 2 ⎟ \n",
" ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 120 \n",
"\n",
" βŽ› \n",
" ________⎞⎞ 2 ⎜ βŽ› 2 \n",
"β•± 2 ⎟⎟ 5 βŽ› u ⎞ ⎜ ⎜ u 1 \n",
" 1 - u ⎠⎟ Ξ”t β‹…βŽœ- ─────────── + λ⎟ β‹…βŽœΞ»β‹…sin(t) + ⎜- ─────────── - ──────────\n",
" ⎟ ⎜ ________ ⎟ ⎜ ⎜ 3/2 _______\n",
" ⎟ ⎜ β•± 2 ⎟ ⎜ ⎜ βŽ› 2⎞ β•± 2\n",
" ⎠ ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ β•²β•± 1 - u \n",
"─────────── + ────────────────────────────────────────────────────────────────\n",
" 120 \n",
"\n",
" 2⎞ \n",
" ⎞ βŽ› ________⎞ ⎟ βŽ› 2 \n",
" ⎟ ⎜ β•± 2 ⎟ ⎟ 5 βŽ› u ⎞ ⎜ u \n",
"β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ 7β‹…Ξ”t β‹…βŽœ- ─────────── + Ξ»βŽŸβ‹…βŽœ- ───────────\n",
"_⎟ ⎟ ⎜ ________ ⎟ ⎜ 3/2\n",
" ⎟ ⎟ ⎜ β•± 2 ⎟ ⎜ βŽ› 2⎞ \n",
" ⎠ ⎠ ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝1 - u ⎠ \n",
"─────────────────────────────────── + ────────────────────────────────────────\n",
" \n",
"\n",
" \n",
" ⎞ βŽ› ________⎞ βŽ› \n",
" 1 ⎟ ⎜ β•± 2 ⎟ ⎜ βŽ› u \n",
" - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u βŽ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»\n",
" ________⎟ ⎜ ⎜ ________ \n",
" β•± 2 ⎟ ⎜ ⎜ β•± 2 \n",
" β•²β•± 1 - u ⎠ ⎝ ⎝ β•²β•± 1 - u \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 120 \n",
"\n",
" βŽ› \n",
" βŽ› ________⎞⎞ ⎜ βŽ› \n",
"⎞ ⎜ β•± 2 ⎟⎟ 5 βŽ› u ⎞ ⎜ ⎜ \n",
"βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ Ξ”t β‹…βŽœ- ─────────── + Ξ»βŽŸβ‹…βŽœΞ»β‹…cos(t) + ⎜- ───\n",
"⎟ ⎟ ⎜ ________ ⎟ ⎜ ⎜ \n",
"⎟ ⎟ ⎜ β•± 2 ⎟ ⎜ ⎜ βŽ› \n",
"⎠ ⎠ ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ ⎝1 \n",
"───────────────────────────────── + ──────────────────────────────────────────\n",
" \n",
"\n",
" 3⎞ \n",
" 3 ⎞ βŽ› ________⎞ ⎟ βŽ› 2 \n",
" 3β‹…u 3β‹…u ⎟ ⎜ β•± 2 ⎟ ⎟ 5 ⎜ u \n",
"──────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ Ξ”t β‹…βŽœ- ─────────── \n",
" 5/2 3/2⎟ ⎟ ⎜ 3/2 \n",
" 2⎞ βŽ› 2⎞ ⎟ ⎟ ⎜ βŽ› 2⎞ \n",
"- u ⎠ ⎝1 - u ⎠ ⎠ ⎠ ⎝ ⎝1 - u ⎠ \n",
"──────────────────────────────────────────────────────── + ───────────────────\n",
" 120 \n",
"\n",
" βŽ› \n",
" ⎞ βŽ› ________⎞ ⎜ βŽ› 2 \n",
" 1 ⎟ ⎜ β•± 2 ⎟ ⎜ ⎜ u \n",
"- β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u βŽ β‹…βŽœΞ»β‹…sin(t) + ⎜- ─────────── - ───\n",
" ________⎟ ⎜ ⎜ 3/2 \n",
" β•± 2 ⎟ ⎜ ⎜ βŽ› 2⎞ β•±\n",
" β•²β•± 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ β•²β•± \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 30 \n",
"\n",
" 2⎞ \n",
" ⎞ βŽ› ________⎞ ⎟ βŽ› 2 ⎞\n",
" 1 ⎟ ⎜ β•± 2 ⎟ ⎟ 5 ⎜ u 1 ⎟\n",
"β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ Ξ”t β‹…βŽœ- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸ\n",
"________⎟ ⎟ ⎜ 3/2 ________⎟\n",
" 2 ⎟ ⎟ ⎜ βŽ› 2⎞ β•± 2 ⎟\n",
" 1 - u ⎠ ⎠ ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠\n",
"────────────────────────────────────────── + ─────────────────────────────────\n",
" \n",
"\n",
" 2 \n",
" βŽ› βŽ› ________⎞⎞ βŽ› \n",
" ⎜ βŽ› u ⎞ ⎜ β•± 2 ⎟⎟ 5 ⎜ \n",
"β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ Ξ”t β‹…βŽœ- ──\n",
" ⎜ ⎜ ________ ⎟ ⎟ ⎜ \n",
" ⎜ ⎜ β•± 2 ⎟ ⎟ ⎜ βŽ› \n",
" ⎝ ⎝ β•²β•± 1 - u ⎠ ⎠ ⎝ ⎝1\n",
"────────────────────────────────────────────────────────────────── + ─────────\n",
" 40 \n",
"\n",
" 2 \n",
" 3 ⎞ βŽ› ________⎞ βŽ› \n",
" 3β‹…u 3β‹…u ⎟ ⎜ β•± 2 ⎟ ⎜ βŽ› \n",
"───────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────\n",
" 5/2 3/2⎟ ⎜ ⎜ __\n",
" 2⎞ βŽ› 2⎞ ⎟ ⎜ ⎜ β•± \n",
" - u ⎠ ⎝1 - u ⎠ ⎠ ⎝ ⎝ β•²β•± 1\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 20 \n",
"\n",
" βŽ› \n",
" βŽ› ________⎞⎞ ⎜ βŽ› \n",
"u ⎞ ⎜ β•± 2 ⎟⎟ 5 ⎜ ⎜ \n",
"────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ Ξ”t β‹…βŽœ-Ξ»β‹…sin(t) + βŽΞ»β‹…(u - sin(t))\n",
"______ ⎟ ⎟ ⎜ \n",
" 2 ⎟ ⎟ ⎜ \n",
" - u ⎠ ⎠ ⎝ \n",
"─────────────────────────────────────────── + ────────────────────────────────\n",
" \n",
"\n",
" 4 ⎞ \n",
" ________⎞ βŽ› 4 2 ⎞⎟ \n",
" β•± 2 ⎟ ⎜ 15β‹…u 18β‹…u 3 ⎟⎟ 4 βŽ› u \n",
" + β•²β•± 1 - u ⎠ β‹…βŽœ- ─────────── - ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβŽŸ Ξ”t β‹…βŽœ- ───────\n",
" ⎜ 7/2 5/2 3/2⎟⎟ ⎜ ____\n",
" ⎜ βŽ› 2⎞ βŽ› 2⎞ βŽ› 2⎞ ⎟⎟ ⎜ β•± \n",
" ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠⎠ ⎝ β•²β•± 1 -\n",
"───────────────────────────────────────────────────────────── + ──────────────\n",
" 120 \n",
"\n",
" \n",
" 2 βŽ› βŽ› ________⎞⎞ \n",
" ⎞ ⎜ βŽ› u ⎞ ⎜ β•± 2 ⎟⎟ \n",
"──── + λ⎟ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ \n",
"____ ⎟ ⎜ ⎜ ________ ⎟ ⎟ \n",
" 2 ⎟ ⎜ ⎜ β•± 2 ⎟ ⎟ \n",
" u ⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ ⎠ \n",
"─────────────────────────────────────────────────────────────────────────── + \n",
" 24 \n",
"\n",
" βŽ› \n",
" ⎜ βŽ› 2 ⎞ βŽ› \n",
" 4 βŽ› u ⎞ ⎜ ⎜ u 1 ⎟ ⎜ \n",
"Ξ”t β‹…βŽœ- ─────────── + Ξ»βŽŸβ‹…βŽœΞ»β‹…sin(t) + ⎜- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(\n",
" ⎜ ________ ⎟ ⎜ ⎜ 3/2 ________⎟ \n",
" ⎜ β•± 2 ⎟ ⎜ ⎜ βŽ› 2⎞ β•± 2 ⎟ \n",
" ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 24 \n",
"\n",
" 2⎞ \n",
" ________⎞ ⎟ βŽ› 2 ⎞ βŽ› \n",
" β•± 2 ⎟ ⎟ 4 ⎜ u 1 ⎟ ⎜ β•±\n",
"t)) + β•²β•± 1 - u ⎠ ⎟ Ξ”t β‹…βŽœ- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± \n",
" ⎟ ⎜ 3/2 ________⎟ \n",
" ⎟ ⎜ βŽ› 2⎞ β•± 2 ⎟ \n",
" ⎠ ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠ \n",
"──────────────────── + ───────────────────────────────────────────────────────\n",
" \n",
"\n",
" \n",
"________⎞ βŽ› βŽ› ________⎞⎞ \n",
" 2 ⎟ ⎜ βŽ› u ⎞ ⎜ β•± 2 ⎟⎟ \n",
" 1 - u βŽ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ Ξ”\n",
" ⎜ ⎜ ________ ⎟ ⎟ \n",
" ⎜ ⎜ β•± 2 ⎟ ⎟ \n",
" ⎝ ⎝ β•²β•± 1 - u ⎠ ⎠ \n",
"────────────────────────────────────────────────────────────────────────── + ─\n",
" 8 \n",
"\n",
" βŽ› 3⎞ \n",
" ⎜ βŽ› 3 ⎞ βŽ› ________⎞ ⎟ \n",
" 4 ⎜ ⎜ 3β‹…u 3β‹…u ⎟ ⎜ β•± 2 ⎟ ⎟ \n",
"t β‹…βŽœΞ»β‹…cos(t) + ⎜- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ \n",
" ⎜ ⎜ 5/2 3/2⎟ ⎟ \n",
" ⎜ ⎜ βŽ› 2⎞ βŽ› 2⎞ ⎟ ⎟ \n",
" ⎝ ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠ ⎠ \n",
"───────────────────────────────────────────────────────────────────────────── \n",
" 24 \n",
"\n",
" \n",
" βŽ› βŽ› \n",
" 3 βŽ› u ⎞ ⎜ βŽ› u ⎞ ⎜ \n",
" Ξ”t β‹…βŽœ- ─────────── + Ξ»βŽŸβ‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²\n",
" ⎜ ________ ⎟ ⎜ ⎜ ________ ⎟ \n",
" ⎜ β•± 2 ⎟ ⎜ ⎜ β•± 2 ⎟ \n",
" ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ \n",
"+ ────────────────────────────────────────────────────────────────────────────\n",
" 6 \n",
"\n",
" βŽ› \n",
" ________⎞⎞ ⎜ βŽ› 2 ⎞ βŽ› \n",
" β•± 2 ⎟⎟ 3 ⎜ ⎜ u 1 ⎟ ⎜ \n",
"β•± 1 - u ⎠⎟ Ξ”t β‹…βŽœΞ»β‹…sin(t) + ⎜- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) +\n",
" ⎟ ⎜ ⎜ 3/2 ________⎟ \n",
" ⎟ ⎜ ⎜ βŽ› 2⎞ β•± 2 ⎟ \n",
" ⎠ ⎝ ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠ \n",
"──────────── + ───────────────────────────────────────────────────────────────\n",
" 6 \n",
"\n",
" 2⎞ \n",
" ________⎞ ⎟ βŽ› βŽ› __\n",
" β•± 2 ⎟ ⎟ 2 ⎜ βŽ› u ⎞ ⎜ β•± \n",
" β•²β•± 1 - u ⎠ ⎟ Ξ”t β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1\n",
" ⎟ ⎜ ⎜ ________ ⎟ \n",
" ⎟ ⎜ ⎜ β•± 2 ⎟ \n",
" ⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ \n",
"─────────────── + ────────────────────────────────────────────────────────────\n",
" 2 \n",
"\n",
" \n",
"______⎞⎞ \n",
" 2 ⎟⎟ \n",
" - u ⎠⎟ \n",
" ⎟ \n",
" ⎟ βŽ› ________⎞ \n",
" ⎠ ⎜ β•± 2 ⎟ \n",
"──────── + Ξ”tβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ + 1\n",
" "
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"evaluate(f_sym,u_sym,Ξ”t,coeffs_ex)[1]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"And their difference, which is the local error:"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\begin{equation*}\\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{3} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{80} + \\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right)^{2} \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{80} - \\frac{105199674209 Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{248351754240} + \\frac{Ξ”t^{5} \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3}\\right)}{80} - \\frac{3844175612059 Ξ”t^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\sin{\\left(t \\right)} + \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)}{6953849118720} + \\frac{902475437790052823 Ξ”t^{5} \\left(- \\frac{u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}} - \\frac{1}{\\sqrt{1 - u^{2}}}\\right) \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2}}{42730909364772720000} + \\frac{1886254549 Ξ”t^{5} \\left(- \\frac{3 u^{3}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3 u}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(- Ξ» \\cos{\\left(t \\right)} + \\left(- \\frac{u}{\\sqrt{1 - u^{2}}} + Ξ»\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)}{40242182400} + \\frac{Ξ”t^{5} \\left(- Ξ» \\sin{\\left(t \\right)} + \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(- \\frac{15 u^{4}}{\\left(1 - u^{2}\\right)^{\\frac{7}{2}}} - \\frac{18 u^{2}}{\\left(1 - u^{2}\\right)^{\\frac{5}{2}}} - \\frac{3}{\\left(1 - u^{2}\\right)^{\\frac{3}{2}}}\\right)\\right)}{80}\\end{equation*}$\n"
],
"text/plain": [
" \n",
" 3 βŽ› βŽ› \n",
" 5 βŽ› u ⎞ ⎜ βŽ› u ⎞ ⎜ \n",
"Ξ”t β‹…βŽœ- ─────────── + λ⎟ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•±\n",
" ⎜ ________ ⎟ ⎜ ⎜ ________ ⎟ \n",
" ⎜ β•± 2 ⎟ ⎜ ⎜ β•± 2 ⎟ \n",
" ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 80 \n",
"\n",
" βŽ› \n",
" ________⎞⎞ 2 ⎜ βŽ› 2 \n",
"β•± 2 ⎟⎟ 5 βŽ› u ⎞ ⎜ ⎜ u 1 \n",
" 1 - u ⎠⎟ Ξ”t β‹…βŽœ- ─────────── + λ⎟ β‹…βŽœΞ»β‹…sin(t) + ⎜- ─────────── - ──────────\n",
" ⎟ ⎜ ________ ⎟ ⎜ ⎜ 3/2 _______\n",
" ⎟ ⎜ β•± 2 ⎟ ⎜ ⎜ βŽ› 2⎞ β•± 2\n",
" ⎠ ⎝ β•²β•± 1 - u ⎠ ⎝ ⎝ ⎝1 - u ⎠ β•²β•± 1 - u \n",
"─────────── + ────────────────────────────────────────────────────────────────\n",
" 80 \n",
"\n",
" 2⎞ \n",
" ⎞ βŽ› ________⎞ ⎟ βŽ› \n",
" ⎟ ⎜ β•± 2 ⎟ ⎟ 5 βŽ› u ⎞ ⎜ \n",
"β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ 105199674209β‹…Ξ”t β‹…βŽœ- ─────────── + Ξ»βŽŸβ‹…βŽœ- \n",
"_⎟ ⎟ ⎜ ________ ⎟ ⎜ \n",
" ⎟ ⎟ ⎜ β•± 2 ⎟ ⎜ \n",
" ⎠ ⎠ ⎝ β•²β•± 1 - u ⎠ ⎝ \n",
"─────────────────────────────────── - ────────────────────────────────────────\n",
" \n",
"\n",
" \n",
" 2 ⎞ βŽ› ________⎞ βŽ› \n",
" u 1 ⎟ ⎜ β•± 2 ⎟ ⎜ βŽ› \n",
"─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u βŽ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ────\n",
" 3/2 ________⎟ ⎜ ⎜ _\n",
"βŽ› 2⎞ β•± 2 ⎟ ⎜ ⎜ β•± \n",
"⎝1 - u ⎠ β•²β•± 1 - u ⎠ ⎝ ⎝ β•²β•± \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 248351754240 \n",
"\n",
" βŽ› \n",
" βŽ› ________⎞⎞ ⎜ \n",
" u ⎞ ⎜ β•± 2 ⎟⎟ 5 βŽ› u ⎞ ⎜ \n",
"─────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ Ξ”t β‹…βŽœ- ─────────── + Ξ»βŽŸβ‹…βŽœΞ»β‹…cos(\n",
"_______ ⎟ ⎟ ⎜ ________ ⎟ ⎜ \n",
" 2 ⎟ ⎟ ⎜ β•± 2 ⎟ ⎜ \n",
"1 - u ⎠ ⎠ ⎝ β•²β•± 1 - u ⎠ ⎝ \n",
"──────────────────────────────────────────── + ───────────────────────────────\n",
" \n",
"\n",
" 3⎞ \n",
" βŽ› 3 ⎞ βŽ› ________⎞ ⎟ \n",
" ⎜ 3β‹…u 3β‹…u ⎟ ⎜ β•± 2 ⎟ ⎟ \n",
"t) + ⎜- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ 38441756\n",
" ⎜ 5/2 3/2⎟ ⎟ \n",
" ⎜ βŽ› 2⎞ βŽ› 2⎞ ⎟ ⎟ \n",
" ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠ ⎠ \n",
"─────────────────────────────────────────────────────────────────── - ────────\n",
" 80 \n",
"\n",
" βŽ› \n",
" βŽ› 2 ⎞ βŽ› ________⎞ ⎜ \n",
" 5 ⎜ u 1 ⎟ ⎜ β•± 2 ⎟ ⎜ \n",
"12059β‹…Ξ”t β‹…βŽœ- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u βŽ β‹…βŽœΞ»β‹…sin(\n",
" ⎜ 3/2 ________⎟ ⎜ \n",
" ⎜ βŽ› 2⎞ β•± 2 ⎟ ⎜ \n",
" ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠ ⎝ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 6953849118720 \n",
"\n",
" 2⎞ \n",
" βŽ› 2 ⎞ βŽ› ________⎞ ⎟ \n",
" ⎜ u 1 ⎟ ⎜ β•± 2 ⎟ ⎟ \n",
"t) + ⎜- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎟ 90247543\n",
" ⎜ 3/2 ________⎟ ⎟ \n",
" ⎜ βŽ› 2⎞ β•± 2 ⎟ ⎟ \n",
" ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠ ⎠ \n",
"─────────────────────────────────────────────────────────────────── + ────────\n",
" \n",
"\n",
" \n",
" βŽ› 2 ⎞ βŽ› \n",
" 5 ⎜ u 1 ⎟ ⎜ βŽ› u ⎞ \n",
"7790052823β‹…Ξ”t β‹…βŽœ- ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβ‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…\n",
" ⎜ 3/2 ________⎟ ⎜ ⎜ ________ ⎟ \n",
" ⎜ βŽ› 2⎞ β•± 2 ⎟ ⎜ ⎜ β•± 2 ⎟ \n",
" ⎝ ⎝1 - u ⎠ β•²β•± 1 - u ⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 42730909364772720000 \n",
"\n",
" 2 \n",
"βŽ› ________⎞⎞ βŽ› 3 \n",
"⎜ β•± 2 ⎟⎟ 5 ⎜ 3β‹…u 3β‹…u \n",
"βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎟ 1886254549β‹…Ξ”t β‹…βŽœ- ─────────── - ───────────\n",
" ⎟ ⎜ 5/2 3/2\n",
" ⎟ ⎜ βŽ› 2⎞ βŽ› 2⎞ \n",
" ⎠ ⎝ ⎝1 - u ⎠ ⎝1 - u ⎠ \n",
"──────────────────────────────── + ───────────────────────────────────────────\n",
" \n",
"\n",
" 2 \n",
"⎞ βŽ› ________⎞ βŽ› βŽ› \n",
"⎟ ⎜ β•± 2 ⎟ ⎜ βŽ› u ⎞ ⎜ \n",
"βŽŸβ‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ β‹…βŽœ-Ξ»β‹…cos(t) + ⎜- ─────────── + Ξ»βŽŸβ‹…βŽΞ»β‹…(u - sin\n",
"⎟ ⎜ ⎜ ________ ⎟ \n",
"⎟ ⎜ ⎜ β•± 2 ⎟ \n",
"⎠ ⎝ ⎝ β•²β•± 1 - u ⎠ \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 40242182400 \n",
"\n",
" βŽ› 4 \n",
" ________⎞⎞ ⎜ βŽ› ________⎞ βŽ› \n",
" β•± 2 ⎟⎟ 5 ⎜ ⎜ β•± 2 ⎟ ⎜ \n",
"(t)) + β•²β•± 1 - u ⎠⎟ Ξ”t β‹…βŽœ-Ξ»β‹…sin(t) + βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ β‹…βŽœ- ───\n",
" ⎟ ⎜ ⎜ \n",
" ⎟ ⎜ ⎜ βŽ› \n",
" ⎠ ⎝ ⎝ ⎝1 \n",
"──────────────────── + ───────────────────────────────────────────────────────\n",
" 80 \n",
"\n",
" ⎞\n",
" 4 2 ⎞⎟\n",
"15β‹…u 18β‹…u 3 ⎟⎟\n",
"──────── - ─────────── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸβŽŸ\n",
" 7/2 5/2 3/2⎟⎟\n",
" 2⎞ βŽ› 2⎞ βŽ› 2⎞ ⎟⎟\n",
"- u ⎠ ⎝1 - u ⎠ ⎝1 - u ⎠ ⎠⎠\n",
"──────────────────────────────────────\n",
" "
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"expr = simplify(evaluate(f_sym,u_sym,Ξ”t,coeffs)-evaluate(f_sym,u_sym,Ξ”t,coeffs_ex))[1]"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\begin{equation*}\\frac{4535459777238976256125 \\left(1 - u^{2}\\right)^{\\frac{35}{2}} \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\left(1 - u^{2}\\right)^{2} \\sin{\\left(t \\right)} - \\sqrt{1 - u^{2}} \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right) + 611621992800 \\left(1 - u^{2}\\right)^{\\frac{33}{2}} \\left(1886254549 u \\left(1 - u^{2}\\right)^{\\frac{3}{2}} \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2} \\left(Ξ» \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right) + 167675760 \\left(- u + Ξ» \\sqrt{1 - u^{2}}\\right) \\left(- 3 u \\left(1 - u^{2}\\right)^{\\frac{3}{2}} \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{3} + Ξ» \\left(1 - u^{2}\\right)^{4} \\cos{\\left(t \\right)}\\right)\\right) - 102554182475454528000 \\left(1 - u^{2}\\right)^{\\frac{27}{2}} \\left(Ξ» \\left(1 - u^{2}\\right)^{\\frac{15}{2}} \\sin{\\left(t \\right)} + 3 \\left(1 - u^{2}\\right)^{4} \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{4} \\left(5 u^{4} + 6 u^{2} \\left(1 - u^{2}\\right) + \\left(1 - u^{2}\\right)^{2}\\right)\\right) + 102554182475454528000 \\left(1 - u^{2}\\right)^{19} \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right)^{3} \\left(Ξ» \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right) + 4 \\left(1 - u^{2}\\right)^{18} \\left(868821451912298236625 \\sqrt{1 - u^{2}} \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right) - 43318821013922535504 \\sqrt{1 - u^{2}} \\left(Ξ» \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)} + \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right) \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)\\right)^{2} + 25638545618863632000 \\left(u - Ξ» \\sqrt{1 - u^{2}}\\right)^{2} \\left(Ξ» \\left(1 - u^{2}\\right)^{2} \\sin{\\left(t \\right)} - \\sqrt{1 - u^{2}} \\left(Ξ» \\left(u - \\sin{\\left(t \\right)}\\right) + \\sqrt{1 - u^{2}}\\right)^{2}\\right)\\right)}{8204334598036362240000 \\left(1 - u^{2}\\right)^{21}}\\end{equation*}$\n"
],
"text/plain": [
" βŽ› \n",
" 35/2 βŽ› ________⎞ ⎜ \n",
" βŽ› 2⎞ ⎜ β•± 2 ⎟ ⎜ βŽ› 2⎞\n",
"4535459777238976256125β‹…βŽ1 - u ⎠ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u βŽ β‹…βŽΞ»β‹…βŽ1 - u ⎠\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" 2⎞ \n",
"2 ________ βŽ› ________⎞ ⎟ \n",
" β•± 2 ⎜ β•± 2 ⎟ ⎟ βŽ› 2\n",
" β‹…sin(t) - β•²β•± 1 - u β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎠ + 611621992800β‹…βŽ1 - u \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" βŽ› 2 \n",
" 33/2 ⎜ 3/2 βŽ› ________⎞ βŽ› ________\n",
"⎞ ⎜ βŽ› 2⎞ ⎜ β•± 2 ⎟ ⎜ β•± 2 \n",
"⎠ β‹…βŽ1886254549β‹…uβ‹…βŽ1 - u ⎠ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ β‹…βŽΞ»β‹…β•²β•± 1 - u \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" \n",
" βŽ› ________⎞ βŽ› ________⎞⎞ βŽ› \n",
" ⎜ β•± 2 ⎟ ⎜ β•± 2 ⎟⎟ ⎜ \n",
"β‹…cos(t) + ⎝u - Ξ»β‹…β•²β•± 1 - u βŽ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎠ + 167675760β‹…βŽ-u \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" βŽ› 3 \n",
" ________⎞ ⎜ 3/2 βŽ› ________⎞ \n",
" β•± 2 ⎟ ⎜ βŽ› 2⎞ ⎜ β•± 2 ⎟ βŽ› \n",
"+ Ξ»β‹…β•²β•± 1 - u βŽ β‹…βŽ- 3β‹…uβ‹…βŽ1 - u ⎠ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ + Ξ»β‹…βŽ1 - u\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" ⎞⎞ βŽ› \n",
" 4 ⎟⎟ 27/2 ⎜ 15/2 \n",
"2⎞ ⎟⎟ βŽ› 2⎞ ⎜ βŽ› 2⎞ βŽ›\n",
" ⎠ β‹…cos(t)⎠⎠ - 102554182475454528000β‹…βŽ1 - u ⎠ β‹…βŽΞ»β‹…βŽ1 - u ⎠ β‹…sin(t) + 3β‹…βŽ\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" 4 ⎞ \n",
" 4 βŽ› ________⎞ βŽ› 2⎞⎟ \n",
" 2⎞ ⎜ β•± 2 ⎟ ⎜ 4 2 βŽ› 2⎞ βŽ› 2⎞ ⎟⎟ \n",
"1 - u ⎠ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ β‹…βŽ5β‹…u + 6β‹…u β‹…βŽ1 - u ⎠ + ⎝1 - u ⎠ ⎠⎠ +\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" βŽ›\n",
" 8204334598036362240000β‹…βŽ\n",
"\n",
" 3 \n",
" 19 βŽ› ________⎞ βŽ› ________ \n",
" βŽ› 2⎞ ⎜ β•± 2 ⎟ ⎜ β•± 2 \n",
" 102554182475454528000β‹…βŽ1 - u ⎠ β‹…βŽu - Ξ»β‹…β•²β•± 1 - u ⎠ β‹…βŽΞ»β‹…β•²β•± 1 - u β‹…cos(t) +\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" 21 \n",
" 2⎞ \n",
"1 - u ⎠ \n",
"\n",
" βŽ› \n",
" βŽ› ________⎞ βŽ› ________⎞⎞ 18 ⎜ \n",
" ⎜ β•± 2 ⎟ ⎜ β•± 2 ⎟⎟ βŽ› 2⎞ ⎜ \n",
" ⎝u - Ξ»β‹…β•²β•± 1 - u βŽ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎠ + 4β‹…βŽ1 - u ⎠ β‹…βŽ868821451\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" \n",
" ________ βŽ› ________⎞ βŽ› ________⎞ βŽ› \n",
" β•± 2 ⎜ β•± 2 ⎟ ⎜ β•± 2 ⎟ ⎜ \n",
"912298236625β‹…β•²β•± 1 - u β‹…βŽu - Ξ»β‹…β•²β•± 1 - u βŽ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u βŽ β‹…βŽΞ»\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" \n",
" ________ βŽ› ________⎞ βŽ› ________⎞⎞ \n",
" β•± 2 ⎜ β•± 2 ⎟ ⎜ β•± 2 ⎟⎟ \n",
"β‹…β•²β•± 1 - u β‹…cos(t) + ⎝u - Ξ»β‹…β•²β•± 1 - u βŽ β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠⎠ - 43\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" \n",
" ________ βŽ› ________ βŽ› ________⎞ βŽ› \n",
" β•± 2 ⎜ β•± 2 ⎜ β•± 2 ⎟ ⎜ \n",
"318821013922535504β‹…β•²β•± 1 - u β‹…βŽΞ»β‹…β•²β•± 1 - u β‹…cos(t) + ⎝u - Ξ»β‹…β•²β•± 1 - u βŽ β‹…βŽΞ»β‹…\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" 2 2 βŽ› \n",
" ________⎞⎞ βŽ› ________⎞ ⎜ \n",
" β•± 2 ⎟⎟ ⎜ β•± 2 ⎟ ⎜ βŽ›\n",
"(u - sin(t)) + β•²β•± 1 - u ⎠⎠ + 25638545618863632000β‹…βŽu - Ξ»β‹…β•²β•± 1 - u ⎠ β‹…βŽΞ»β‹…βŽ\n",
"──────────────────────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" \n",
"\n",
" 2⎞⎞\n",
" 2 ________ βŽ› ________⎞ ⎟⎟\n",
" 2⎞ β•± 2 ⎜ β•± 2 ⎟ ⎟⎟\n",
"1 - u ⎠ β‹…sin(t) - β•²β•± 1 - u β‹…βŽΞ»β‹…(u - sin(t)) + β•²β•± 1 - u ⎠ ⎠⎠\n",
"───────────────────────────────────────────────────────────────\n",
" \n",
" \n",
" "
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"sp.simplify(expand(expr))"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\begin{equation*}- \\frac{3 u^{8} Ξ”t^{5}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{6} Ξ”t^{5} \\sqrt{1 - u^{2}}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3 u^{6} Ξ”t^{5}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{6} Ξ”t^{5}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} + \\frac{u^{6} Ξ”t^{5}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} - \\frac{5407445509 u^{6} Ξ”t^{5}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3 u^{4} Ξ”t^{5}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{4} Ξ”t^{5}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} - \\frac{u^{4} Ξ”t^{5}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} + \\frac{8425609189 u^{4} Ξ”t^{5}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{328105563238046242399 u^{4} Ξ”t^{5}}{241303958765775360000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{2} Ξ”t^{5}}{40 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3635043306271280591533 u^{2} Ξ”t^{5}}{4102167299018181120000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{754418498879 u^{2} Ξ”t^{5}}{772649902080 \\sqrt{1 - u^{2}}} - \\frac{3 Ξ”t^{5}}{80 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 Ξ”t^{5}}{6953849118720 \\sqrt{1 - u^{2}}} + Ξ»^{5} \\left(\\frac{u Ξ”t^{5}}{80} - \\frac{Ξ”t^{5} \\sin{\\left(t \\right)}}{80}\\right) + Ξ»^{4} \\left(- \\frac{3 u^{8} Ξ”t^{5}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3 u^{7} Ξ”t^{5} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{6} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{5407445509 u^{6} Ξ”t^{5}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3 u^{5} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{6413500069 u^{5} Ξ”t^{5} \\sin{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3 u^{4} Ξ”t^{5} \\sin^{4}{\\left(t \\right)}}{16 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{8425609189 u^{4} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{357614120019468965321 u^{4} Ξ”t^{5}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{14461936549 u^{3} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{98031318961151965421 u^{3} Ξ”t^{5} \\sin{\\left(t \\right)}}{1025541824754545280000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{2} Ξ”t^{5} \\sin^{4}{\\left(t \\right)}}{40 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{757632721070389930279 u^{2} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{697309902804057541121 u^{2} Ξ”t^{5}}{2051083649509090560000 \\sqrt{1 - u^{2}}} + \\frac{4401390949 u Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{748586994041784805121 u Ξ”t^{5} \\sin{\\left(t \\right)}}{1025541824754545280000 \\sqrt{1 - u^{2}}} + \\frac{Ξ”t^{5} \\sqrt{1 - u^{2}}}{80} - \\frac{Ξ”t^{5} \\cos{\\left(t \\right)}}{80} - \\frac{3 Ξ”t^{5} \\sin^{4}{\\left(t \\right)}}{80 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{799864085279512069121 Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}}\\right) + Ξ»^{3} \\left(- \\frac{3 u^{7} Ξ”t^{5} \\sqrt{1 - u^{2}}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{7} Ξ”t^{5}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3 u^{7} Ξ”t^{5}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} + \\frac{1886254549 u^{7} Ξ”t^{5}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} + \\frac{9 u^{6} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{6} Ξ”t^{5} \\sin{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{9 u^{6} Ξ”t^{5} \\sin{\\left(t \\right)}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} - \\frac{5658763647 u^{6} Ξ”t^{5} \\sin{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{9 u^{5} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{6413500069 u^{5} Ξ”t^{5} \\sqrt{1 - u^{2}}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{5} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{9 u^{5} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} + \\frac{5658763647 u^{5} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} + \\frac{1886254549 u^{5} Ξ”t^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{5} Ξ”t^{5}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{16341676594871 u^{5} Ξ”t^{5}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{105199674209 u^{5} Ξ”t^{5}}{124175877120 u^{4} - 248351754240 u^{2} + 124175877120} + \\frac{3 u^{5} Ξ”t^{5}}{80 u^{4} - 160 u^{2} + 80} + \\frac{3 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{3}{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{8425609189 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{2235676800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{4} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{6953849118720 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3 u^{4} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{- 80 u^{6} + 240 u^{4} - 240 u^{2} + 80} - \\frac{1886254549 u^{4} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{1886254549 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{10682727341193180000 u^{4} - 21365454682386360000 u^{2} + 10682727341193180000} + \\frac{49459645354533 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} + \\frac{105199674209 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{62087938560 u^{4} - 124175877120 u^{2} + 62087938560} - \\frac{9 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{80 u^{4} - 160 u^{2} + 80} - \\frac{14461936549 u^{3} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{98031318961151965421 u^{3} Ξ”t^{5} \\sqrt{1 - u^{2}}}{1025541824754545280000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{1886254549 u^{3} Ξ”t^{5} \\sin^{2}{\\left(t \\right)} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{3} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{49894260924453 u^{3} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{105199674209 u^{3} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{124175877120 u^{4} - 248351754240 u^{2} + 124175877120} + \\frac{9 u^{3} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{80 u^{4} - 160 u^{2} + 80} - \\frac{493765118337591853817 u^{3} Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{10319047364622345641093 u^{3} Ξ”t^{5}}{8204334598036362240000 \\left(1 - u^{2}\\right)} + \\frac{9 u^{2} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{3}{\\left(t \\right)}}{10 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{757632721070389930279 u^{2} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{1025541824754545280000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{16776292164791 u^{2} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{3 u^{2} Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{80 u^{4} - 160 u^{2} + 80} + \\frac{12079201575925326001 u^{2} Ξ”t^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{120651979382887680000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{25788879601336394706311 u^{2} Ξ”t^{5} \\sin{\\left(t \\right)}}{8204334598036362240000 \\left(1 - u^{2}\\right)} - \\frac{4401390949 u Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{748586994041784805121 u Ξ”t^{5}}{1025541824754545280000} + \\frac{1886254549 u Ξ”t^{5} \\sin^{2}{\\left(t \\right)} \\cos{\\left(t \\right)}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{20005292013953025321343 u Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{8204334598036362240000 \\left(1 - u^{2}\\right)} - \\frac{705268173027862269617 u Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}} + \\frac{3 Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{3}{\\left(t \\right)}}{20 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{787044812470080253121 Ξ”t^{5} \\sin{\\left(t \\right)}}{1025541824754545280000} + \\frac{3844175612059 Ξ”t^{5} \\sin^{3}{\\left(t \\right)}}{6953849118720 \\left(1 - u^{2}\\right)} + \\frac{782183809884453165617 Ξ”t^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}}\\right) + Ξ»^{2} \\left(\\frac{9 u^{8} Ξ”t^{5}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{7} Ξ”t^{5} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{6} Ξ”t^{5} \\sqrt{1 - u^{2}}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{9 u^{6} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{6} Ξ”t^{5}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{6} Ξ”t^{5}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} + \\frac{105199674209 u^{6} Ξ”t^{5}}{248351754240 u^{4} \\sqrt{1 - u^{2}} - 496703508480 u^{2} \\sqrt{1 - u^{2}} + 248351754240 \\sqrt{1 - u^{2}}} - \\frac{u^{6} Ξ”t^{5}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} + \\frac{5407445509 u^{6} Ξ”t^{5}}{2235676800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{5} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{1158974853120 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{9 u^{5} Ξ”t^{5} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{5} Ξ”t^{5} \\sin{\\left(t \\right)}}{21365454682386360000 u^{4} \\sqrt{1 - u^{2}} - 42730909364772720000 u^{2} \\sqrt{1 - u^{2}} + 21365454682386360000 \\sqrt{1 - u^{2}}} - \\frac{105199674209 u^{5} Ξ”t^{5} \\sin{\\left(t \\right)}}{124175877120 u^{4} \\sqrt{1 - u^{2}} - 248351754240 u^{2} \\sqrt{1 - u^{2}} + 124175877120 \\sqrt{1 - u^{2}}} + \\frac{2 u^{5} Ξ”t^{5} \\sin{\\left(t \\right)}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} - \\frac{6413500069 u^{5} Ξ”t^{5} \\sin{\\left(t \\right)}}{1490451200 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin^{2}{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{1886254549 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{9 u^{4} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{8 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{4} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{42730909364772720000 u^{4} \\sqrt{1 - u^{2}} - 85461818729545440000 u^{2} \\sqrt{1 - u^{2}} + 42730909364772720000 \\sqrt{1 - u^{2}}} + \\frac{105199674209 u^{4} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{248351754240 u^{4} \\sqrt{1 - u^{2}} - 496703508480 u^{2} \\sqrt{1 - u^{2}} + 248351754240 \\sqrt{1 - u^{2}}} - \\frac{u^{4} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{80 u^{4} \\sqrt{1 - u^{2}} - 160 u^{2} \\sqrt{1 - u^{2}} + 80 \\sqrt{1 - u^{2}}} + \\frac{8425609189 u^{4} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{8425609189 u^{4} Ξ”t^{5}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{782183809884453165617 u^{4} Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 u^{4} - 4102167299018181120000 u^{2} + 2051083649509090560000} - \\frac{14903400795783417131743 u^{4} Ξ”t^{5}}{4102167299018181120000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{1886254549 u^{3} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{14461936549 u^{3} Ξ”t^{5} \\sin{\\left(t \\right)}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{782183809884453165617 u^{3} Ξ”t^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{2051083649509090560000 u^{4} - 4102167299018181120000 u^{2} + 2051083649509090560000} + \\frac{48926120124603963505793 u^{3} Ξ”t^{5} \\sin{\\left(t \\right)}}{8204334598036362240000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{12079201575925326001 u^{2} Ξ”t^{5} \\sqrt{1 - u^{2}} \\cos{\\left(t \\right)}}{120651979382887680000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{27 u^{2} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{20 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{22318774874155177518791 u^{2} Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{8204334598036362240000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{2} Ξ”t^{5} \\cos^{2}{\\left(t \\right)}}{42730909364772720000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{757632721070389930279 u^{2} Ξ”t^{5}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{705268173027862269617 u^{2} Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(1 - u^{2}\\right)} - \\frac{5102610026961050007259 u^{2} Ξ”t^{5}}{1640866919607272448000 \\sqrt{1 - u^{2}}} - \\frac{1886254549 u Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{6707030400 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{4401390949 u Ξ”t^{5} \\sin{\\left(t \\right)}}{4471353600 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{782183809884453165617 u Ξ”t^{5} \\sin{\\left(t \\right)} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(1 - u^{2}\\right)} + \\frac{14780311813398039280937 u Ξ”t^{5} \\sin{\\left(t \\right)}}{2734778199345454080000 \\sqrt{1 - u^{2}}} + \\frac{105199674209 Ξ”t^{5} \\sqrt{1 - u^{2}}}{248351754240} - \\frac{756545264265589533617 Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000} - \\frac{9 Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{40 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 Ξ”t^{5} \\sin^{2}{\\left(t \\right)}}{1738462279680 \\sqrt{1 - u^{2}}} - \\frac{902475437790052823 Ξ”t^{5} \\cos^{2}{\\left(t \\right)}}{42730909364772720000 \\sqrt{1 - u^{2}}} - \\frac{1436611804849711823 Ξ”t^{5}}{42730909364772720000 \\sqrt{1 - u^{2}}}\\right) + Ξ» \\left(\\frac{3 u^{7} Ξ”t^{5} \\sqrt{1 - u^{2}}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{7} Ξ”t^{5}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3395336389 u^{7} Ξ”t^{5}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{3 u^{6} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{6} Ξ”t^{5} \\sin{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3395336389 u^{6} Ξ”t^{5} \\sin{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{3 u^{5} Ξ”t^{5} \\sqrt{1 - u^{2}}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{6078148549 u^{5} Ξ”t^{5} \\sqrt{1 - u^{2}}}{4471353600 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{3844175612059 u^{5} Ξ”t^{5}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} + \\frac{3395336389 u^{5} Ξ”t^{5}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} - \\frac{1886254549 u^{5} Ξ”t^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{902475437790052823 u^{5} Ξ”t^{5}}{10682727341193180000 u^{4} - 21365454682386360000 u^{2} + 10682727341193180000} + \\frac{3186582816619 u^{5} Ξ”t^{5}}{1931624755200 u^{4} - 3863249510400 u^{2} + 1931624755200} + \\frac{u^{5} Ξ”t^{5}}{80 \\left(u^{4} - 2 u^{2} + 1\\right)} + \\frac{3 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{4 \\left(- u^{6} \\sqrt{1 - u^{2}} + 3 u^{4} \\sqrt{1 - u^{2}} - 3 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{7922581909 u^{4} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{6707030400 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{3844175612059 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{2317949706240 \\left(- u^{6} + 3 u^{4} - 3 u^{2} + 1\\right)} - \\frac{3395336389 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{- 13414060800 u^{6} + 40242182400 u^{4} - 40242182400 u^{2} + 13414060800} + \\frac{902475437790052823 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{3234873435499 u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{1931624755200 u^{4} - 3863249510400 u^{2} + 1931624755200} - \\frac{u^{4} Ξ”t^{5} \\sin{\\left(t \\right)}}{80 \\left(u^{4} - 2 u^{2} + 1\\right)} - \\frac{14461936549 u^{3} Ξ”t^{5} \\sqrt{1 - u^{2}}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{22917976411669 u^{3} Ξ”t^{5} \\sqrt{1 - u^{2}}}{8692311398400 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{1886254549 u^{3} Ξ”t^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{902475437790052823 u^{3} Ξ”t^{5}}{21365454682386360000 u^{4} - 42730909364772720000 u^{2} + 21365454682386360000} - \\frac{14385906530231 u^{3} Ξ”t^{5}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} + \\frac{519403663956455485817 u^{3} Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{9972496796510965357061 u^{3} Ξ”t^{5}}{8204334598036362240000 \\left(1 - u^{2}\\right)} + \\frac{9 u^{2} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{10 \\left(u^{4} \\sqrt{1 - u^{2}} - 2 u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{86326046719459 u^{2} Ξ”t^{5} \\sqrt{1 - u^{2}} \\sin{\\left(t \\right)}}{34769245593600 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} + \\frac{14820522100151 u^{2} Ξ”t^{5} \\sin{\\left(t \\right)}}{17384622796800 u^{4} - 34769245593600 u^{2} + 17384622796800} - \\frac{10831818276999618281093 u^{2} Ξ”t^{5} \\sin{\\left(t \\right)}}{8204334598036362240000 \\left(1 - u^{2}\\right)} - \\frac{1886254549 u Ξ”t^{5} \\sqrt{1 - u^{2}}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{826854427199 u Ξ”t^{5}}{386324951040} + \\frac{1886254549 u Ξ”t^{5} \\cos{\\left(t \\right)}}{13414060800 \\left(- u^{2} \\sqrt{1 - u^{2}} + \\sqrt{1 - u^{2}}\\right)} - \\frac{234341226128040406829 u Ξ”t^{5}}{482607917531550720000 \\left(1 - u^{2}\\right)} + \\frac{756545264265589533617 u Ξ”t^{5} \\cos{\\left(t \\right)}}{2051083649509090560000 \\sqrt{1 - u^{2}}} + \\frac{4162893696667 Ξ”t^{5} \\sin{\\left(t \\right)}}{2317949706240} + \\frac{3844175612059 Ξ”t^{5} \\sin{\\left(t \\right)}}{6953849118720 \\left(1 - u^{2}\\right)}\\right)\\end{equation*}$\n"
],
"text/plain": [
" \n",
" 8 5 \n",
" 3β‹…u β‹…Ξ”t \n",
"- ───────────────────────────────────────────────────────────────────────── + \n",
" βŽ› ________ ________ ________ ________⎞ \n",
" ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 β•± 2 ⎟ \n",
" 16β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" ________ \n",
" 6 5 β•± 2 6 \n",
" 3844175612059β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u 3β‹…u β‹…\n",
"────────────────────────────────────── + ─────────────────────────────────────\n",
" βŽ› 6 4 2 ⎞ βŽ› ________ _______\n",
"6953849118720β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ ⎜ 6 β•± 2 4 β•± 2\n",
" 8β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u \n",
"\n",
" \n",
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"Ξ”t \n",
"─────────────────────────────────── + ────────────────────────────────────────\n",
"_ ________ ________⎞ ________ \n",
" 2 β•± 2 β•± 2 ⎟ 4 β•± 2 \n",
" - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 42730909364772720000β‹…u β‹…β•²β•± 1 - u - 85\n",
"\n",
" \n",
" 6 5 \n",
" 902475437790052823β‹…u β‹…Ξ”t \n",
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"461818729545440000β‹…u β‹…β•²β•± 1 - u + 42730909364772720000β‹…β•²β•± 1 - u 80β‹…u β‹…β•²\n",
"\n",
" \n",
" 6 5 \n",
" u β‹…Ξ”t 54074\n",
"──────────────────────────────────────────────── - ───────────────────────────\n",
" ________ ________ ________ βŽ› ________\n",
" β•± 2 2 β•± 2 β•± 2 ⎜ 4 β•± 2 \n",
"β•± 1 - u - 160β‹…u β‹…β•²β•± 1 - u + 80β‹…β•²β•± 1 - u 13414060800β‹…βŽu β‹…β•²β•± 1 - u \n",
"\n",
" ________ \n",
" 6 5 4 5 β•± 2 \n",
"45509β‹…u β‹…Ξ”t 3844175612059β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u \n",
"────────────────────────────────── - ────────────────────────────────────── - \n",
" ________ ________⎞ βŽ› 6 4 2 ⎞ \n",
" 2 β•± 2 β•± 2 ⎟ 6953849118720β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ \n",
" - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 4 5 \n",
" 3β‹…u β‹…Ξ”t \n",
"───────────────────────────────────────────────────────────────────────── - ──\n",
" βŽ› ________ ________ ________ ________⎞ \n",
" ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 β•± 2 ⎟ \n",
"16β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 42\n",
"\n",
" \n",
" 4 5 \n",
" 902475437790052823β‹…u β‹…Ξ”t \n",
"──────────────────────────────────────────────────────────────────────────────\n",
" ________ ________ \n",
" 4 β•± 2 2 β•± 2 \n",
"730909364772720000β‹…u β‹…β•²β•± 1 - u - 85461818729545440000β‹…u β‹…β•²β•± 1 - u + 4273\n",
"\n",
" \n",
" 4 5 \n",
" u β‹…Ξ”t \n",
"──────────────────────────── - ───────────────────────────────────────────────\n",
" ________ ________ ________ \n",
" β•± 2 4 β•± 2 2 β•± 2 β•±\n",
"0909364772720000β‹…β•²β•± 1 - u 80β‹…u β‹…β•²β•± 1 - u - 160β‹…u β‹…β•²β•± 1 - u + 80β‹…β•²β•± \n",
"\n",
" \n",
" 4 5 \n",
" 8425609189β‹…u β‹…Ξ”t \n",
"──────── + ───────────────────────────────────────────────────────────── + ───\n",
"________ βŽ› ________ ________ ________⎞ \n",
" 2 ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ \n",
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"\n",
" \n",
" 4 5 \n",
" 328105563238046242399β‹…u β‹…Ξ”t 9β‹…\n",
"─────────────────────────────────────────────────── - ────────────────────────\n",
" βŽ› ________ ________⎞ βŽ› ________ \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± 2 \n",
"303958765775360000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 40β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u\n",
"\n",
" \n",
" 2 5 2 5 \n",
"u β‹…Ξ”t 3635043306271280591533β‹…u β‹…Ξ”t \n",
"──────────────────────────── - ───────────────────────────────────────────────\n",
" ________ ________⎞ βŽ› ________ _\n",
"2 β•± 2 β•± 2 ⎟ ⎜ 2 β•± 2 β•± \n",
" β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 4102167299018181120000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± \n",
"\n",
" \n",
" 2 5 5 \n",
" 754418498879β‹…u β‹…Ξ”t 3β‹…Ξ”t \n",
"──────── + ──────────────────────── - ─────────────────────────────────── - ──\n",
"_______⎞ ________ βŽ› ________ ________⎞ \n",
" 2 ⎟ β•± 2 ⎜ 2 β•± 2 β•± 2 ⎟ \n",
"1 - u ⎠ 772649902080β‹…β•²β•± 1 - u 80β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 69\n",
"\n",
" βŽ› \n",
" 5 βŽ› 5 5 ⎞ ⎜ \n",
" 3844175612059β‹…Ξ”t 5 ⎜uβ‹…Ξ”t Ξ”t β‹…sin(t)⎟ 4 ⎜ \n",
"─────────────────────── + Ξ» β‹…βŽœβ”€β”€β”€β”€β”€ - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸ + Ξ» β‹…βŽœ- ────────────────────\n",
" ________ ⎝ 80 80 ⎠ ⎜ βŽ› ________\n",
" β•± 2 ⎜ ⎜ 6 β•± 2 \n",
"53849118720β‹…β•²β•± 1 - u ⎝ 16β‹…βŽ- u β‹…β•²β•± 1 - u \n",
"\n",
" \n",
" 8 5 \n",
" 3β‹…u β‹…Ξ”t \n",
"───────────────────────────────────────────────────── + ──────────────────────\n",
" ________ ________ ________⎞ βŽ› ________ \n",
" 4 β•± 2 2 β•± 2 β•± 2 ⎟ ⎜ 6 β•± 2 \n",
" + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 4β‹…βŽ- u β‹…β•²β•± 1 - u + \n",
"\n",
" \n",
" 7 5 \n",
" 3β‹…u β‹…Ξ”t β‹…sin(t) \n",
"────────────────────────────────────────────────── - ─────────────────────────\n",
" ________ ________ ________⎞ βŽ› ________ \n",
" 4 β•± 2 2 β•± 2 β•± 2 ⎟ ⎜ 6 β•± 2 \n",
"3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 8β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u\n",
"\n",
" \n",
" 6 5 2 \n",
" 9β‹…u β‹…Ξ”t β‹…sin (t) 540744\n",
"─────────────────────────────────────────────── - ────────────────────────────\n",
" ________ ________ ________⎞ βŽ› ________ \n",
"4 β•± 2 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± 2 \n",
" β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 13414060800β‹…βŽu β‹…β•²β•± 1 - u \n",
"\n",
" \n",
" 6 5 5 5 3 \n",
"5509β‹…u β‹…Ξ”t 3β‹…u β‹…Ξ”t β‹…sin (\n",
"───────────────────────────────── + ──────────────────────────────────────────\n",
" ________ ________⎞ βŽ› ________ ________ \n",
" 2 β•± 2 β•± 2 ⎟ ⎜ 6 β•± 2 4 β•± 2 \n",
"- 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3\n",
"\n",
" \n",
" 5 5 \n",
"t) 6413500069β‹…u β‹…Ξ”t β‹…sin(t) \n",
"────────────────────────────── + ─────────────────────────────────────────────\n",
" ________ ________⎞ βŽ› ________ ________\n",
" 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 \n",
"β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u \n",
"\n",
" \n",
" 4 5 4 \n",
" 3β‹…u β‹…Ξ”t β‹…sin (t) \n",
"─────────────── - ────────────────────────────────────────────────────────────\n",
" ________⎞ βŽ› ________ ________ ________ \n",
" β•± 2 ⎟ ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 \n",
" + β•²β•± 1 - u ⎠ 16β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u +\n",
"\n",
" \n",
" 4 5 2 \n",
" 8425609189β‹…u β‹…Ξ”t β‹…sin (t) \n",
"───────────── - ──────────────────────────────────────────────────────────── +\n",
" ________⎞ βŽ› ________ ________ ________⎞ \n",
" β•± 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ \n",
" β•²β•± 1 - u ⎠ 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 4 5 \n",
" 357614120019468965321β‹…u β‹…Ξ”t 1\n",
" ─────────────────────────────────────────────────────── + ───────────────────\n",
" βŽ› ________ ________⎞ βŽ› \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•±\n",
" 2051083649509090560000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 13414060800β‹…βŽu β‹…β•²β•± \n",
"\n",
" \n",
" 3 5 3 \n",
"4461936549β‹…u β‹…Ξ”t β‹…sin (t) 98031318961151965421β‹…u\n",
"────────────────────────────────────────── - ─────────────────────────────────\n",
"________ ________ ________⎞ βŽ› _\n",
" 2 2 β•± 2 β•± 2 ⎟ ⎜ 2 β•± \n",
" 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 1025541824754545280000β‹…βŽ- u β‹…β•²β•± \n",
"\n",
" \n",
"3 5 2 5 4 \n",
" β‹…Ξ”t β‹…sin(t) 9β‹…u β‹…Ξ”t β‹…sin (t) \n",
"────────────────────── - ──────────────────────────────────────────────────── \n",
"_______ ________⎞ βŽ› ________ ________ ________⎞ \n",
" 2 β•± 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ \n",
"1 - u + β•²β•± 1 - u ⎠ 40β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 2 5 2 \n",
" 757632721070389930279β‹…u β‹…Ξ”t β‹…sin (t) 697309902804057\n",
"- ─────────────────────────────────────────────────────── + ──────────────────\n",
" βŽ› ________ ________⎞ \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ \n",
" 2051083649509090560000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 205108364950909056\n",
"\n",
" \n",
" 2 5 5 3 \n",
"541121β‹…u β‹…Ξ”t 4401390949β‹…uβ‹…Ξ”t β‹…sin (t) 748586994041\n",
"──────────────── + ──────────────────────────────────────────── - ────────────\n",
" ________ βŽ› ________ ________⎞ \n",
" β•± 2 ⎜ 2 β•± 2 β•± 2 ⎟ \n",
"0000β‹…β•²β•± 1 - u 13414060800β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 102554182475\n",
"\n",
" ________ \n",
" 5 5 β•± 2 5 5 4 \n",
"784805121β‹…uβ‹…Ξ”t β‹…sin(t) Ξ”t β‹…β•²β•± 1 - u Ξ”t β‹…cos(t) 3β‹…Ξ”t β‹…sin (\n",
"────────────────────── + ─────────────── - ────────── - ──────────────────────\n",
" ________ 80 80 βŽ› ________ \n",
" β•± 2 ⎜ 2 β•± 2 \n",
"4545280000β‹…β•²β•± 1 - u 80β‹…βŽ- u β‹…β•²β•± 1 - u +\n",
"\n",
" ⎞ βŽ› \n",
" 5 2 ⎟ ⎜ \n",
"t) 799864085279512069121β‹…Ξ”t β‹…sin (t) ⎟ 3 ⎜ \n",
"───────────── + β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸ + Ξ» β‹…βŽœ- ──────────────────\n",
" ________⎞ ________⎟ ⎜ βŽ› _______\n",
" β•± 2 ⎟ β•± 2 ⎟ ⎜ ⎜ 6 β•± 2\n",
" β•²β•± 1 - u ⎠ 2051083649509090560000β‹…β•²β•± 1 - u ⎠ ⎝ 4β‹…βŽ- u β‹…β•²β•± 1 - u \n",
"\n",
" ________ \n",
" 7 5 β•± 2 \n",
" 3β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u 384417561205\n",
"────────────────────────────────────────────────────── - ─────────────────────\n",
"_ ________ ________ ________⎞ βŽ› 6 \n",
" 4 β•± 2 2 β•± 2 β•± 2 ⎟ 6953849118720β‹…βŽ- u +\n",
" + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 7 5 7 5 \n",
"9β‹…u β‹…Ξ”t 3β‹…u β‹…Ξ”t 18\n",
"───────────────── + ────────────────────────────── + ─────────────────────────\n",
" 4 2 ⎞ 6 4 2 6 \n",
" 3β‹…u - 3β‹…u + 1⎠ - 80β‹…u + 240β‹…u - 240β‹…u + 80 - 13414060800β‹…u + 402421\n",
" \n",
"\n",
" __\n",
" 7 5 6 5 β•± \n",
"86254549β‹…u β‹…Ξ”t 9β‹…u β‹…Ξ”t β‹…β•²β•± 1\n",
"─────────────────────────────────────── + ────────────────────────────────────\n",
" 4 2 βŽ› ________ ______\n",
"82400β‹…u - 40242182400β‹…u + 13414060800 ⎜ 6 β•± 2 4 β•± \n",
" 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u\n",
"\n",
"______ \n",
" 2 6 5 \n",
" - u β‹…sin(t) 3844175612059β‹…u β‹…Ξ”t β‹…sin(t) \n",
"──────────────────────────────────── + ────────────────────────────────────── \n",
"__ ________ ________⎞ βŽ› 6 4 2 ⎞ \n",
"2 2 β•± 2 β•± 2 ⎟ 2317949706240β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ \n",
" - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 6 5 6 5 \n",
" 9β‹…u β‹…Ξ”t β‹…sin(t) 5658763647β‹…u β‹…Ξ”t β‹…sin(t\n",
"- ────────────────────────────── - ───────────────────────────────────────────\n",
" 6 4 2 6 4 \n",
" - 80β‹…u + 240β‹…u - 240β‹…u + 80 - 13414060800β‹…u + 40242182400β‹…u - 4024218\n",
" \n",
"\n",
" ________ \n",
" 5 5 β•± 2 2 \n",
") 9β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin (t) \n",
"───────────────────── - ──────────────────────────────────────────────────────\n",
" 2 βŽ› ________ ________ _____\n",
"2400β‹…u + 13414060800 ⎜ 6 β•± 2 4 β•± 2 2 β•± \n",
" 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - \n",
"\n",
" ________ \n",
" 5 5 β•± 2 \n",
" 6413500069β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u \n",
"────────────────── - ─────────────────────────────────────────────────────────\n",
"___ ________⎞ βŽ› ________ ________ ______\n",
" 2 β•± 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± \n",
"u + β•²β•± 1 - u ⎠ 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u\n",
"\n",
" \n",
" 5 5 2 5 5 2 \n",
" 3844175612059β‹…u β‹…Ξ”t β‹…sin (t) 9β‹…u β‹…Ξ”t β‹…sin (t) \n",
"─── - ────────────────────────────────────── + ────────────────────────────── \n",
"__⎞ βŽ› 6 4 2 ⎞ 6 4 2 \n",
"2 ⎟ 2317949706240β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ - 80β‹…u + 240β‹…u - 240β‹…u + 80 \n",
" ⎠ \n",
"\n",
" \n",
" 5 5 2 \n",
" 5658763647β‹…u β‹…Ξ”t β‹…sin (t) \n",
"+ ──────────────────────────────────────────────────────────────── + ─────────\n",
" 6 4 2 \n",
" - 13414060800β‹…u + 40242182400β‹…u - 40242182400β‹…u + 13414060800 \n",
" 134140608\n",
"\n",
" \n",
" 5 5 \n",
" 1886254549β‹…u β‹…Ξ”t β‹…cos(t) \n",
"──────────────────────────────────────────────────── + ───────────────────────\n",
" βŽ› ________ ________ ________⎞ 4\n",
" ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ 21365454682386360000β‹…u \n",
"00β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 5 5 \n",
"902475437790052823β‹…u β‹…Ξ”t 1634167659\n",
"───────────────────────────────────────────────── - ──────────────────────────\n",
" 2 4 \n",
" - 42730909364772720000β‹…u + 21365454682386360000 17384622796800β‹…u - 347692\n",
" \n",
"\n",
" \n",
" 5 5 5 5 \n",
"4871β‹…u β‹…Ξ”t 105199674209β‹…u β‹…Ξ”t \n",
"──────────────────────────── - ───────────────────────────────────────────────\n",
" 2 4 2 \n",
"45593600β‹…u + 17384622796800 124175877120β‹…u - 248351754240β‹…u + 12417587712\n",
" \n",
"\n",
" ________ \n",
" 5 5 4 5 β•± 2 3 \n",
" 3β‹…u β‹…Ξ”t 3β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin (t) \n",
"─ + ─────────────────── + ────────────────────────────────────────────────────\n",
" 4 2 βŽ› ________ ________ ___\n",
"0 80β‹…u - 160β‹…u + 80 ⎜ 6 β•± 2 4 β•± 2 2 β•± \n",
" 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 \n",
"\n",
" ________ \n",
" 4 5 β•± 2 \n",
" 8425609189β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) \n",
"──────────────────── + ───────────────────────────────────────────────────────\n",
"_____ ________⎞ βŽ› ________ ________ ____\n",
" 2 β•± 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± \n",
"- u + β•²β•± 1 - u ⎠ 2235676800β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 -\n",
"\n",
" \n",
" 4 5 3 4 5 3 \n",
" 3844175612059β‹…u β‹…Ξ”t β‹…sin (t) 3β‹…u β‹…Ξ”t β‹…sin (t) \n",
"───── + ────────────────────────────────────── - ─────────────────────────────\n",
"____⎞ βŽ› 6 4 2 ⎞ 6 4 2 \n",
" 2 ⎟ 6953849118720β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ - 80β‹…u + 240β‹…u - 240β‹…u + 8\n",
" u ⎠ \n",
"\n",
" \n",
" 4 5 3 \n",
" 1886254549β‹…u β‹…Ξ”t β‹…sin (t) \n",
"─ - ──────────────────────────────────────────────────────────────── - ───────\n",
" 6 4 2 \n",
"0 - 13414060800β‹…u + 40242182400β‹…u - 40242182400β‹…u + 13414060800 \n",
" 6707030\n",
"\n",
" \n",
" 4 5 \n",
" 1886254549β‹…u β‹…Ξ”t β‹…sin(t)β‹…cos(t) 90\n",
"───────────────────────────────────────────────────── - ──────────────────────\n",
" βŽ› ________ ________ ________⎞ \n",
" ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ 10682727341193180000β‹…u\n",
"400β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 4 5 \n",
"2475437790052823β‹…u β‹…Ξ”t β‹…sin(t) 494596453545\n",
"────────────────────────────────────────────────── + ─────────────────────────\n",
"4 2 4 \n",
" - 21365454682386360000β‹…u + 10682727341193180000 17384622796800β‹…u - 34769\n",
" \n",
"\n",
" \n",
" 4 5 4 5 \n",
"33β‹…u β‹…Ξ”t β‹…sin(t) 105199674209β‹…u β‹…Ξ”t β‹…sin(t) \n",
"───────────────────────────── + ──────────────────────────────────────────────\n",
" 2 4 2 \n",
"245593600β‹…u + 17384622796800 62087938560β‹…u - 124175877120β‹…u + 62087938560\n",
" \n",
"\n",
" ________ \n",
" 4 5 3 5 β•± 2 2 \n",
" 9β‹…u β‹…Ξ”t β‹…sin(t) 14461936549β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin (t) \n",
" - ─────────────────── - ─────────────────────────────────────────────────────\n",
" 4 2 βŽ› ________ ________ __\n",
" 80β‹…u - 160β‹…u + 80 ⎜ 4 β•± 2 2 β•± 2 β•± \n",
" 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1\n",
"\n",
" ________ \n",
" 3 5 β•± 2 \n",
" 98031318961151965421β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u \n",
"─────── + ─────────────────────────────────────────────────────── + ──────────\n",
"______⎞ βŽ› ________ ________⎞ \n",
" 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ \n",
" - u ⎠ 1025541824754545280000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 1341406080\n",
"\n",
" \n",
" 3 5 2 \n",
" 1886254549β‹…u β‹…Ξ”t β‹…sin (t)β‹…cos(t) 90247\n",
"─────────────────────────────────────────────────── + ────────────────────────\n",
" βŽ› ________ ________ ________⎞ 4 \n",
" ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ 21365454682386360000β‹…u \n",
"0β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 3 5 2 \n",
"5437790052823β‹…u β‹…Ξ”t β‹…sin (t) 49894260924453β‹…\n",
"──────────────────────────────────────────────── - ───────────────────────────\n",
" 2 4 \n",
"- 42730909364772720000β‹…u + 21365454682386360000 17384622796800β‹…u - 3476924\n",
" \n",
"\n",
" \n",
" 3 5 2 3 5 2 \n",
"u β‹…Ξ”t β‹…sin (t) 105199674209β‹…u β‹…Ξ”t β‹…sin (t) \n",
"─────────────────────────── - ────────────────────────────────────────────────\n",
" 2 4 2 \n",
"5593600β‹…u + 17384622796800 124175877120β‹…u - 248351754240β‹…u + 124175877120\n",
" \n",
"\n",
" \n",
" 3 5 2 3 5 \n",
" 9β‹…u β‹…Ξ”t β‹…sin (t) 493765118337591853817β‹…u β‹…Ξ”t β‹…cos(t) \n",
" + ─────────────────── - ─────────────────────────────────────────────────────\n",
" 4 2 βŽ› ________ _______\n",
" 80β‹…u - 160β‹…u + 80 ⎜ 2 β•± 2 β•± 2\n",
" 2051083649509090560000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u \n",
"\n",
" ________ \n",
" 3 5 2 5 β•± 2 3 \n",
" 10319047364622345641093β‹…u β‹…Ξ”t 9β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin (t\n",
"── - ─────────────────────────────── + ───────────────────────────────────────\n",
"_⎞ βŽ› 2⎞ βŽ› ________ ________ \n",
" ⎟ 8204334598036362240000β‹…βŽ1 - u ⎠ ⎜ 4 β•± 2 2 β•± 2 \n",
" ⎠ 10β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u +\n",
"\n",
" ________ \n",
" 2 5 β•± 2 \n",
") 757632721070389930279β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) \n",
"───────────── + ─────────────────────────────────────────────────────── + ────\n",
" ________⎞ βŽ› ________ ________⎞ \n",
" β•± 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ 1738\n",
" β•²β•± 1 - u ⎠ 1025541824754545280000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 2 5 3 2 5 3 \n",
" 16776292164791β‹…u β‹…Ξ”t β‹…sin (t) 3β‹…u β‹…Ξ”t β‹…sin (t) \n",
"────────────────────────────────────────────────── - ─────────────────── + ───\n",
" 4 2 4 2 \n",
"4622796800β‹…u - 34769245593600β‹…u + 17384622796800 80β‹…u - 160β‹…u + 80 \n",
" 120\n",
"\n",
" \n",
" 2 5 \n",
" 12079201575925326001β‹…u β‹…Ξ”t β‹…sin(t)β‹…cos(t) 25788879601336394706311β‹…\n",
"─────────────────────────────────────────────────── + ────────────────────────\n",
" βŽ› ________ ________⎞ \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ 820433459803636224000\n",
"651979382887680000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" ________ \n",
" 2 5 5 β•± 2 2 \n",
"u β‹…Ξ”t β‹…sin(t) 4401390949β‹…uβ‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin (t) 7485869940417848\n",
"───────────── - ─────────────────────────────────────────── + ────────────────\n",
" βŽ› 2⎞ βŽ› ________ ________⎞ 1025541824754\n",
"0β‹…βŽ1 - u ⎠ ⎜ 2 β•± 2 β•± 2 ⎟ \n",
" 4471353600β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 5 5 2 \n",
"05121β‹…uβ‹…Ξ”t 1886254549β‹…uβ‹…Ξ”t β‹…sin (t)β‹…cos(t) 20005292013953025\n",
"─────────── + ──────────────────────────────────────────── - ─────────────────\n",
"545280000 βŽ› ________ ________⎞ \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ 82043345980363\n",
" 13414060800β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" ____\n",
" 5 2 5 5 β•± \n",
"321343β‹…uβ‹…Ξ”t β‹…sin (t) 705268173027862269617β‹…uβ‹…Ξ”t β‹…cos(t) 3β‹…Ξ”t β‹…β•²β•± 1 -\n",
"──────────────────── - ────────────────────────────────── + ──────────────────\n",
" βŽ› 2⎞ ________ βŽ› ______\n",
"62240000β‹…βŽ1 - u ⎠ β•± 2 ⎜ 2 β•± \n",
" 2051083649509090560000β‹…β•²β•± 1 - u 20β‹…βŽ- u β‹…β•²β•± 1 - u\n",
"\n",
"____ \n",
" 2 3 5 5 3 \n",
" u β‹…sin (t) 787044812470080253121β‹…Ξ”t β‹…sin(t) 3844175612059β‹…Ξ”t β‹…sin (\n",
"───────────────── - ──────────────────────────────── + ───────────────────────\n",
"__ ________⎞ 1025541824754545280000 βŽ› 2\n",
"2 β•± 2 ⎟ 6953849118720β‹…βŽ1 - u \n",
" + β•²β•± 1 - u ⎠ \n",
"\n",
" ⎞ βŽ› \n",
" 5 ⎟ ⎜ \n",
"t) 782183809884453165617β‹…Ξ”t β‹…sin(t)β‹…cos(t)⎟ 2 ⎜ \n",
"── + β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸ + Ξ» β‹…βŽœβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€\n",
"⎞ ________ ⎟ ⎜ βŽ› ________ \n",
"⎠ β•± 2 ⎟ ⎜ ⎜ 6 β•± 2 4\n",
" 2051083649509090560000β‹…β•²β•± 1 - u ⎠ ⎝8β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u \n",
"\n",
" \n",
" 8 5 \n",
" 9β‹…u β‹…Ξ”t 9\n",
"────────────────────────────────────────────── - ─────────────────────────────\n",
" ________ ________ ________⎞ βŽ› ________ \n",
" β•± 2 2 β•± 2 β•± 2 ⎟ ⎜ 6 β•± 2 4 \n",
"β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•±\n",
"\n",
" _____\n",
" 7 5 6 5 β•± \n",
"β‹…u β‹…Ξ”t β‹…sin(t) 3844175612059β‹…u β‹…Ξ”t β‹…β•²β•± 1 - \n",
"─────────────────────────────────────────── - ────────────────────────────────\n",
" ________ ________ ________⎞ βŽ› 6 4 \n",
"β•± 2 2 β•± 2 β•± 2 ⎟ 2317949706240β‹…βŽ- u + 3β‹…u - 3β‹…u\n",
" 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
"___ \n",
" 2 6 5 2 \n",
"u 9β‹…u β‹…Ξ”t β‹…sin (t) \n",
"────── + ─────────────────────────────────────────────────────────────────────\n",
"2 ⎞ βŽ› ________ ________ ________ ______\n",
" + 1⎠ ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 β•± \n",
" 8β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u\n",
"\n",
" \n",
" 6 5 \n",
" 9β‹…u β‹…Ξ”t \n",
"─── - ────────────────────────────────────────────────────────────────────────\n",
"__⎞ βŽ› ________ ________ ________ ________⎞\n",
"2 ⎟ ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 β•± 2 ⎟\n",
" ⎠ 8β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠\n",
"\n",
" \n",
" 6 5 \n",
" 902475437790052823β‹…u β‹…Ξ”t \n",
" - ───────────────────────────────────────────────────────────────────────────\n",
" ________ ________ \n",
" 4 β•± 2 2 β•± 2 \n",
" 42730909364772720000β‹…u β‹…β•²β•± 1 - u - 85461818729545440000β‹…u β‹…β•²β•± 1 - u +\n",
"\n",
" \n",
" \n",
" 1051996742\n",
"───────────────────────────────── + ──────────────────────────────────────────\n",
" ________ ________ \n",
" β•± 2 4 β•± 2 \n",
" 42730909364772720000β‹…β•²β•± 1 - u 248351754240β‹…u β‹…β•²β•± 1 - u - 496703508480\n",
"\n",
" \n",
" 6 5 6 5 \n",
"09β‹…u β‹…Ξ”t u β‹…Ξ”t \n",
"────────────────────────────────────────── - ─────────────────────────────────\n",
" ________ ________ ________ ___\n",
" 2 β•± 2 β•± 2 4 β•± 2 2 β•± \n",
"β‹…u β‹…β•²β•± 1 - u + 248351754240β‹…β•²β•± 1 - u 80β‹…u β‹…β•²β•± 1 - u - 160β‹…u β‹…β•²β•± 1 \n",
"\n",
" \n",
" 6 5 \n",
" 5407445509β‹…u β‹…Ξ”t \n",
"────────────────────── + ─────────────────────────────────────────────────────\n",
"_____ ________ βŽ› ________ ________ __\n",
" 2 β•± 2 ⎜ 4 β•± 2 2 β•± 2 β•± \n",
"- u + 80β‹…β•²β•± 1 - u 2235676800β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1\n",
"\n",
" ________ \n",
" 5 5 β•± 2 \n",
" 3844175612059β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) \n",
"─────── + ─────────────────────────────────────── + ──────────────────────────\n",
"______⎞ βŽ› 6 4 2 ⎞ βŽ› ________ \n",
" 2 ⎟ 1158974853120β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ ⎜ 6 β•± 2 4\n",
" - u ⎠ 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u \n",
"\n",
" \n",
" 5 5 \n",
" 9β‹…u β‹…Ξ”t β‹…sin(t) \n",
"────────────────────────────────────────────── + ─────────────────────────────\n",
" ________ ________ ________⎞ __\n",
" β•± 2 2 β•± 2 β•± 2 ⎟ 4 β•± \n",
"β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 21365454682386360000β‹…u β‹…β•²β•± 1\n",
"\n",
" \n",
" 5 5 \n",
" 902475437790052823β‹…u β‹…Ξ”t β‹…sin(t) \n",
"──────────────────────────────────────────────────────────────────────────────\n",
"______ ________ _______\n",
" 2 2 β•± 2 β•± 2\n",
" - u - 42730909364772720000β‹…u β‹…β•²β•± 1 - u + 21365454682386360000β‹…β•²β•± 1 - u \n",
"\n",
" \n",
" 5 5 \n",
" 105199674209β‹…u β‹…Ξ”t β‹…sin(t) \n",
"─ - ──────────────────────────────────────────────────────────────────────────\n",
"_ ________ ________ \n",
" 4 β•± 2 2 β•± 2 \n",
" 124175877120β‹…u β‹…β•²β•± 1 - u - 248351754240β‹…u β‹…β•²β•± 1 - u + 124175877120β‹…β•²\n",
"\n",
" \n",
" 5 5 \n",
" 2β‹…u β‹…Ξ”t β‹…sin(t) \n",
"────────── + ─────────────────────────────────────────────────────── - ───────\n",
" ________ ________ ________ ________ \n",
" β•± 2 4 β•± 2 2 β•± 2 β•± 2 \n",
"β•± 1 - u 80β‹…u β‹…β•²β•± 1 - u - 160β‹…u β‹…β•²β•± 1 - u + 80β‹…β•²β•± 1 - u 1490451\n",
"\n",
" \n",
" 5 5 4 5 \n",
" 6413500069β‹…u β‹…Ξ”t β‹…sin(t) 3844175612059β‹…u β‹…Ξ”t β‹…β•²\n",
"───────────────────────────────────────────────────── - ──────────────────────\n",
" βŽ› ________ ________ ________⎞ βŽ› 6 \n",
" ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ 2317949706240β‹…βŽ- u +\n",
"200β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" ________ ________ \n",
" β•± 2 2 4 5 β•± 2 \n",
"β•± 1 - u β‹…sin (t) 1886254549β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…cos(t) \n",
"────────────────── + ─────────────────────────────────────────────────────────\n",
" 4 2 ⎞ βŽ› ________ ________ ______\n",
" 3β‹…u - 3β‹…u + 1⎠ ⎜ 4 β•± 2 2 β•± 2 β•± \n",
" 6707030400β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u\n",
"\n",
" \n",
" 4 5 2 \n",
" 9β‹…u β‹…Ξ”t β‹…sin (t) \n",
"─── - ────────────────────────────────────────────────────────────────────────\n",
"__⎞ βŽ› ________ ________ ________ ________⎞\n",
"2 ⎟ ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 β•± 2 ⎟\n",
" ⎠ 8β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠\n",
"\n",
" \n",
" 4 5 2 \n",
" 902475437790052823β‹…u β‹…Ξ”t β‹…sin (t) \n",
" - ───────────────────────────────────────────────────────────────────────────\n",
" ________ ________ \n",
" 4 β•± 2 2 β•± 2 \n",
" 42730909364772720000β‹…u β‹…β•²β•± 1 - u - 85461818729545440000β‹…u β‹…β•²β•± 1 - u +\n",
"\n",
" \n",
" \n",
" 105199674209β‹…u\n",
"───────────────────────────────── + ──────────────────────────────────────────\n",
" ________ ________ \n",
" β•± 2 4 β•± 2 \n",
" 42730909364772720000β‹…β•²β•± 1 - u 248351754240β‹…u β‹…β•²β•± 1 - u - 496703508480\n",
"\n",
" \n",
"4 5 2 4 5 2 \n",
" β‹…Ξ”t β‹…sin (t) u β‹…Ξ”t β‹…sin (\n",
"────────────────────────────────────────── - ─────────────────────────────────\n",
" ________ ________ ________ ___\n",
" 2 β•± 2 β•± 2 4 β•± 2 2 β•± \n",
"β‹…u β‹…β•²β•± 1 - u + 248351754240β‹…β•²β•± 1 - u 80β‹…u β‹…β•²β•± 1 - u - 160β‹…u β‹…β•²β•± 1 \n",
"\n",
" \n",
" 4 5 2 \n",
"t) 8425609189β‹…u β‹…Ξ”t β‹…sin (t) \n",
"────────────────────── + ─────────────────────────────────────────────────────\n",
"_____ ________ βŽ› ________ ________ __\n",
" 2 β•± 2 ⎜ 4 β•± 2 2 β•± 2 β•± \n",
"- u + 80β‹…β•²β•± 1 - u 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1\n",
"\n",
" \n",
" 4 5 \n",
" 8425609189β‹…u β‹…Ξ”t \n",
"─────── - ──────────────────────────────────────────────────────────── + ─────\n",
"______⎞ βŽ› ________ ________ ________⎞ \n",
" 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ 20510\n",
" - u ⎠ 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 4 5 \n",
" 782183809884453165617β‹…u β‹…Ξ”t β‹…cos(t) \n",
"───────────────────────────────────────────────────────────────────────── - ──\n",
" 4 2 \n",
"83649509090560000β‹…u - 4102167299018181120000β‹…u + 2051083649509090560000 \n",
" 41\n",
"\n",
" \n",
" 4 5 3 \n",
" 14903400795783417131743β‹…u β‹…Ξ”t 1886254549β‹…u β‹…\n",
"───────────────────────────────────────────────────── - ──────────────────────\n",
" βŽ› ________ ________⎞ βŽ› ____\n",
" ⎜ 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± \n",
"02167299018181120000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 6707030400β‹…βŽu β‹…β•²β•± 1 -\n",
"\n",
" ________ \n",
" 5 β•± 2 3 5 \n",
"Ξ”t β‹…β•²β•± 1 - u β‹…sin(t)β‹…cos(t) 14461936549β‹…u β‹…Ξ”t β‹…s\n",
"────────────────────────────────────── + ─────────────────────────────────────\n",
"____ ________ ________⎞ βŽ› ________ \n",
" 2 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± 2 2 β•±\n",
" u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± \n",
"\n",
" \n",
" 3 5 \n",
"in(t) 782183809884453165617β‹…u β‹…Ξ”t β‹…sin(t\n",
"─────────────────────── - ────────────────────────────────────────────────────\n",
"________ ________⎞ 4 \n",
" 2 β•± 2 ⎟ 2051083649509090560000β‹…u - 4102167299018181120000β‹…u\n",
" 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 3 5 \n",
")β‹…cos(t) 48926120124603963505793β‹…u β‹…Ξ”t β‹…sin(t) \n",
"────────────────────────── + ─────────────────────────────────────────────────\n",
"2 βŽ› ________ ___\n",
" + 2051083649509090560000 ⎜ 2 β•± 2 β•± \n",
" 8204334598036362240000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 \n",
"\n",
" ________ \n",
" 2 5 β•± 2 \n",
" 12079201575925326001β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…cos(t) \n",
"────── - ────────────────────────────────────────────────────── - ────────────\n",
"_____⎞ βŽ› ________ ________⎞ βŽ› __\n",
" 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± \n",
"- u ⎠ 120651979382887680000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 20β‹…βŽu β‹…β•²β•± 1\n",
"\n",
" \n",
" 2 5 2 2\n",
" 27β‹…u β‹…Ξ”t β‹…sin (t) 22318774874155177518791β‹…u \n",
"──────────────────────────────────────── - ───────────────────────────────────\n",
"______ ________ ________⎞ βŽ› ___\n",
" 2 2 β•± 2 β•± 2 ⎟ ⎜ 2 β•± \n",
" - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 8204334598036362240000β‹…βŽ- u β‹…β•²β•± 1 \n",
"\n",
" \n",
" 5 2 2 5 2 \n",
"β‹…Ξ”t β‹…sin (t) 902475437790052823β‹…u β‹…Ξ”t β‹…cos (t) \n",
"──────────────────── - ───────────────────────────────────────────────────── -\n",
"_____ ________⎞ βŽ› ________ ________⎞ \n",
" 2 β•± 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ \n",
"- u + β•²β•± 1 - u ⎠ 42730909364772720000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 2 5 \n",
" 757632721070389930279β‹…u β‹…Ξ”t 7052681730278622696\n",
" ─────────────────────────────────────────────────────── + ───────────────────\n",
" βŽ› ________ ________⎞ \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ 20510836495090905\n",
" 2051083649509090560000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" _\n",
" 2 5 2 5 5 β•± \n",
"17β‹…u β‹…Ξ”t β‹…cos(t) 5102610026961050007259β‹…u β‹…Ξ”t 1886254549β‹…uβ‹…Ξ”t β‹…β•²β•± \n",
"──────────────── - ────────────────────────────────── - ──────────────────────\n",
" βŽ› 2⎞ ________ βŽ› __\n",
"60000β‹…βŽ1 - u ⎠ β•± 2 ⎜ 2 β•± \n",
" 1640866919607272448000β‹…β•²β•± 1 - u 6707030400β‹…βŽ- u β‹…β•²β•± 1\n",
"\n",
"_______ \n",
" 2 5 \n",
"1 - u β‹…sin(t)β‹…cos(t) 4401390949β‹…uβ‹…Ξ”t β‹…sin(t) 78218380\n",
"───────────────────── + ─────────────────────────────────────────── - ────────\n",
"______ ________⎞ βŽ› ________ ________⎞ \n",
" 2 β•± 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ 205\n",
" - u + β•²β•± 1 - u ⎠ 4471353600β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 5 5 \n",
"9884453165617β‹…uβ‹…Ξ”t β‹…sin(t)β‹…cos(t) 14780311813398039280937β‹…uβ‹…Ξ”t β‹…sin(t) 105\n",
"───────────────────────────────── + ──────────────────────────────────── + ───\n",
" βŽ› 2⎞ ________ \n",
"1083649509090560000β‹…βŽ1 - u ⎠ β•± 2 \n",
" 2734778199345454080000β‹…β•²β•± 1 - u \n",
"\n",
" ________ \n",
" 5 β•± 2 5 \n",
"199674209β‹…Ξ”t β‹…β•²β•± 1 - u 756545264265589533617β‹…Ξ”t β‹…cos(t) 9β‹…Ξ”t\n",
"───────────────────────── - ──────────────────────────────── - ───────────────\n",
" 248351754240 2051083649509090560000 βŽ› ___\n",
" ⎜ 2 β•± \n",
" 40β‹…βŽ- u β‹…β•²β•± 1 \n",
"\n",
" \n",
"5 2 5 2 5 \n",
" β‹…sin (t) 3844175612059β‹…Ξ”t β‹…sin (t) 902475437790052823β‹…Ξ”t β‹…cos\n",
"──────────────────── - ───────────────────────── - ───────────────────────────\n",
"_____ ________⎞ ________ ___\n",
" 2 β•± 2 ⎟ β•± 2 β•± \n",
"- u + β•²β•± 1 - u ⎠ 1738462279680β‹…β•²β•± 1 - u 42730909364772720000β‹…β•²β•± 1 \n",
"\n",
" ⎞ βŽ› \n",
"2 5 ⎟ ⎜ 7 \n",
" (t) 1436611804849711823β‹…Ξ”t ⎟ ⎜ 3β‹…u β‹…\n",
"───── - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸ + Ξ»β‹…βŽœβ”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€\n",
"_____ ________⎟ ⎜ βŽ› ________ _\n",
" 2 β•± 2 ⎟ ⎜ ⎜ 6 β•± 2 4 β•± \n",
"- u 42730909364772720000β‹…β•²β•± 1 - u ⎠ ⎝4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± \n",
"\n",
" ________ \n",
" 5 β•± 2 7 5 \n",
"Ξ”t β‹…β•²β•± 1 - u 3844175612059β‹…u β‹…Ξ”t \n",
"───────────────────────────────────────── + ──────────────────────────────────\n",
"_______ ________ ________⎞ βŽ› 6 4 2 \n",
" 2 2 β•± 2 β•± 2 ⎟ 2317949706240β‹…βŽ- u + 3β‹…u - 3β‹…u \n",
"1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 7 5 \n",
" 3395336389β‹…u β‹…Ξ”t \n",
"──── - ──────────────────────────────────────────────────────────────── - ────\n",
" ⎞ 6 4 2 βŽ› \n",
"+ 1⎠ - 13414060800β‹…u + 40242182400β‹…u - 40242182400β‹…u + 13414060800 ⎜ \n",
" 4β‹…βŽ-\n",
"\n",
" ________ \n",
" 6 5 β•± 2 \n",
" 3β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) 38\n",
"──────────────────────────────────────────────────────────────────── - ───────\n",
" ________ ________ ________ ________⎞ \n",
" 6 β•± 2 4 β•± 2 2 β•± 2 β•± 2 ⎟ 2317949\n",
" u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 6 5 6 5 \n",
"44175612059β‹…u β‹…Ξ”t β‹…sin(t) 3395336389β‹…u β‹…Ξ”t β‹…sin(t)\n",
"─────────────────────────────── + ────────────────────────────────────────────\n",
" βŽ› 6 4 2 ⎞ 6 4 \n",
"706240β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ - 13414060800β‹…u + 40242182400β‹…u - 40242182\n",
" \n",
"\n",
" ________ \n",
" 5 5 β•± 2 \n",
" 3β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u \n",
"──────────────────── - ───────────────────────────────────────────────────────\n",
" 2 βŽ› ________ ________ ______\n",
"400β‹…u + 13414060800 ⎜ 6 β•± 2 4 β•± 2 2 β•± \n",
" 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u\n",
"\n",
" ________ \n",
" 5 5 β•± 2 \n",
" 6078148549β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u \n",
"───────────────── + ──────────────────────────────────────────────────────────\n",
"__ ________⎞ βŽ› ________ ________ _______\n",
"2 β•± 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± 2\n",
" + β•²β•± 1 - u ⎠ 4471353600β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u \n",
"\n",
" \n",
" 5 5 \n",
" 3844175612059β‹…u β‹…Ξ”t 339533638\n",
"── - ────────────────────────────────────── + ────────────────────────────────\n",
"_⎞ βŽ› 6 4 2 ⎞ 6 \n",
" ⎟ 2317949706240β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ - 13414060800β‹…u + 40242182400β‹…u\n",
" ⎠ \n",
"\n",
" \n",
" 5 5 5 5 \n",
"9β‹…u β‹…Ξ”t 1886254549β‹…u β‹…Ξ”t β‹…cos(t)\n",
"──────────────────────────────── - ───────────────────────────────────────────\n",
"4 2 βŽ› ________ _____\n",
" - 40242182400β‹…u + 13414060800 ⎜ 4 β•± 2 2 β•± \n",
" 13414060800β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - \n",
"\n",
" \n",
" 5 5 \n",
" 902475437790052823β‹…u β‹…Ξ”t \n",
"────────────────── - ─────────────────────────────────────────────────────────\n",
"___ ________⎞ 4 2 \n",
" 2 β•± 2 ⎟ 10682727341193180000β‹…u - 21365454682386360000β‹…u + 10682\n",
"u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 5 5 \n",
" 3186582816619β‹…u β‹…Ξ”t \n",
"─────────────── + ─────────────────────────────────────────────────── + ──────\n",
" 4 2 βŽ› 4\n",
"727341193180000 1931624755200β‹…u - 3863249510400β‹…u + 1931624755200 80β‹…βŽu \n",
" \n",
"\n",
" ________ \n",
" 5 5 4 5 β•± 2 \n",
"u β‹…Ξ”t 3β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) \n",
"──────────── + ───────────────────────────────────────────────────────────────\n",
" 2 ⎞ βŽ› ________ ________ ________ \n",
" - 2β‹…u + 1⎠ ⎜ 6 β•± 2 4 β•± 2 2 β•± 2 β•±\n",
" 4β‹…βŽ- u β‹…β•²β•± 1 - u + 3β‹…u β‹…β•²β•± 1 - u - 3β‹…u β‹…β•²β•± 1 - u + β•²β•± \n",
"\n",
" ________ \n",
" 4 5 β•± 2 \n",
" 7922581909β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) \n",
"───────── - ──────────────────────────────────────────────────────────── + ───\n",
"________⎞ βŽ› ________ ________ ________⎞ \n",
" 2 ⎟ ⎜ 4 β•± 2 2 β•± 2 β•± 2 ⎟ 231\n",
" 1 - u ⎠ 6707030400β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 4 5 4 5 \n",
" 3844175612059β‹…u β‹…Ξ”t β‹…sin(t) 3395336389β‹…u β‹…Ξ”t β‹…si\n",
"─────────────────────────────────── - ────────────────────────────────────────\n",
" βŽ› 6 4 2 ⎞ 6 4 \n",
"7949706240β‹…βŽ- u + 3β‹…u - 3β‹…u + 1⎠ - 13414060800β‹…u + 40242182400β‹…u - 4024\n",
" \n",
"\n",
" \n",
" 4 5 \n",
"n(t) 902475437790052823β‹…u β‹…Ξ”t β‹…sin(t\n",
"──────────────────────── + ───────────────────────────────────────────────────\n",
" 2 4 2 \n",
"2182400β‹…u + 13414060800 21365454682386360000β‹…u - 42730909364772720000β‹…u +\n",
" \n",
"\n",
" \n",
" 4 5 \n",
") 3234873435499β‹…u β‹…Ξ”t β‹…sin(t) \n",
"───────────────────── - ─────────────────────────────────────────────────── - \n",
" 4 2 \n",
" 21365454682386360000 1931624755200β‹…u - 3863249510400β‹…u + 1931624755200 \n",
" \n",
"\n",
" ________ \n",
" 4 5 3 5 β•± 2 \n",
" u β‹…Ξ”t β‹…sin(t) 14461936549β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u \n",
"────────────────── - ─────────────────────────────────────────────────────────\n",
" βŽ› 4 2 ⎞ βŽ› ________ ________ _____\n",
"80β‹…βŽu - 2β‹…u + 1⎠ ⎜ 4 β•± 2 2 β•± 2 β•± \n",
" 13414060800β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - \n",
"\n",
" ________ \n",
" 3 5 β•± 2 \n",
" 22917976411669β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u 188\n",
"──── - ────────────────────────────────────────────── + ──────────────────────\n",
"___⎞ βŽ› ________ ________⎞ βŽ› ___\n",
" 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ ⎜ 4 β•± \n",
"u ⎠ 8692311398400β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 13414060800β‹…βŽu β‹…β•²β•± 1 \n",
"\n",
" \n",
" 3 5 \n",
"6254549β‹…u β‹…Ξ”t β‹…cos(t) 9024754377900\n",
"─────────────────────────────────────── + ────────────────────────────────────\n",
"_____ ________ ________⎞ 4 \n",
" 2 2 β•± 2 β•± 2 ⎟ 21365454682386360000β‹…u - 4273090936\n",
"- u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 3 5 3 5 \n",
"52823β‹…u β‹…Ξ”t 14385906530231β‹…u β‹…Ξ”t \n",
"──────────────────────────────────── - ───────────────────────────────────────\n",
" 2 4 2 \n",
"4772720000β‹…u + 21365454682386360000 17384622796800β‹…u - 34769245593600β‹…u +\n",
" \n",
"\n",
" \n",
" 3 5 \n",
" 519403663956455485817β‹…u β‹…Ξ”t β‹…cos(t) 9\n",
"─────────────── + ─────────────────────────────────────────────────────── + ──\n",
" βŽ› ________ ________⎞ \n",
" 17384622796800 ⎜ 2 β•± 2 β•± 2 ⎟ 82\n",
" 2051083649509090560000β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" ________ \n",
" 3 5 2 5 β•± 2 \n",
"972496796510965357061β‹…u β‹…Ξ”t 9β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) \n",
"───────────────────────────── + ──────────────────────────────────────────────\n",
" βŽ› 2⎞ βŽ› ________ ________ ___\n",
"04334598036362240000β‹…βŽ1 - u ⎠ ⎜ 4 β•± 2 2 β•± 2 β•± \n",
" 10β‹…βŽu β‹…β•²β•± 1 - u - 2β‹…u β‹…β•²β•± 1 - u + β•²β•± 1 \n",
"\n",
" ________ \n",
" 2 5 β•± 2 \n",
" 86326046719459β‹…u β‹…Ξ”t β‹…β•²β•± 1 - u β‹…sin(t) 148205\n",
"────── + ─────────────────────────────────────────────── + ───────────────────\n",
"_____⎞ βŽ› ________ ________⎞ 4 \n",
" 2 ⎟ ⎜ 2 β•± 2 β•± 2 ⎟ 17384622796800β‹…u -\n",
"- u ⎠ 34769245593600β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ \n",
"\n",
" \n",
" 2 5 2 5 \n",
"22100151β‹…u β‹…Ξ”t β‹…sin(t) 10831818276999618281093β‹…u β‹…Ξ”t β‹…sin(t) \n",
"─────────────────────────────────── - ───────────────────────────────────── - \n",
" 2 βŽ› 2⎞ \n",
" 34769245593600β‹…u + 17384622796800 8204334598036362240000β‹…βŽ1 - u ⎠ \n",
" \n",
"\n",
" ________ \n",
" 5 β•± 2 5 \n",
" 1886254549β‹…uβ‹…Ξ”t β‹…β•²β•± 1 - u 826854427199β‹…uβ‹…Ξ”t \n",
"──────────────────────────────────────────── - ────────────────── + ──────────\n",
" βŽ› ________ ________⎞ 386324951040 \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ \n",
"13414060800β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 1341406080\n",
"\n",
" \n",
" 5 5 \n",
"1886254549β‹…uβ‹…Ξ”t β‹…cos(t) 234341226128040406829β‹…uβ‹…Ξ”t 75654526\n",
"────────────────────────────────── - ────────────────────────────── + ────────\n",
" βŽ› ________ ________⎞ βŽ› 2⎞ \n",
" ⎜ 2 β•± 2 β•± 2 ⎟ 482607917531550720000β‹…βŽ1 - u ⎠ \n",
"0β‹…βŽ- u β‹…β•²β•± 1 - u + β•²β•± 1 - u ⎠ 20510836\n",
"\n",
" \n",
" 5 5 5 \n",
"4265589533617β‹…uβ‹…Ξ”t β‹…cos(t) 4162893696667β‹…Ξ”t β‹…sin(t) 3844175612059β‹…Ξ”t β‹…sin(\n",
"────────────────────────── + ──────────────────────── + ──────────────────────\n",
" ________ 2317949706240 βŽ› 2\n",
" β•± 2 6953849118720β‹…βŽ1 - u \n",
"49509090560000β‹…β•²β•± 1 - u \n",
"\n",
" ⎞\n",
" ⎟\n",
"t)⎟\n",
"β”€β”€βŽŸ\n",
"⎞ ⎟\n",
"⎠ ⎟\n",
" ⎠"
]
},
"execution_count": 38,
"metadata": {},
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}
],
"source": [
"expr2 = sp.collect(sp.expand(expr),Ξ»)"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/latex": [
"$\\begin{equation*}Ξ»^{5} \\left(\\frac{u Ξ”t^{5}}{80} - \\frac{Ξ”t^{5} \\sin{\\left(t \\right)}}{80}\\right)\\end{equation*}$\n"
],
"text/plain": [
" βŽ› 5 5 ⎞\n",
" 5 ⎜uβ‹…Ξ”t Ξ”t β‹…sin(t)⎟\n",
"Ξ» β‹…βŽœβ”€β”€β”€β”€β”€ - β”€β”€β”€β”€β”€β”€β”€β”€β”€β”€βŽŸ\n",
" ⎝ 80 80 ⎠"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"expr2.args[5]"
]
}
],
"metadata": {
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"display_name": "Julia 1.6.3",
"language": "julia",
"name": "julia-1.6"
},
"language_info": {
"file_extension": ".jl",
"mimetype": "application/julia",
"name": "julia",
"version": "1.6.3"
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"toc": {
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