60.3.217 problem 1232
Internal
problem
ID
[11213]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1232
Date
solved
:
Sunday, March 30, 2025 at 07:46:19 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-n \left (n +1\right ) y+\frac {d}{d x}\operatorname {LegendreP}\left (n , x\right )&=0 \end{align*}
✓ Maple. Time used: 0.106 (sec). Leaf size: 409
ode:=(x^2-1)*diff(diff(y(x),x),x)-n*(n+1)*y(x)+Diff(LegendreP(n,x),x) = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 11.047 (sec). Leaf size: 468
ode=(-(n*LegendreP[-1 + n, x]) + n*x*LegendreP[n, x])/(-1 + x^2) - n*(1 + n)*y[x] + (-1 + x^2)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},x^2\right ) \int _1^x\frac {3 n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[1]^2\right ) K[1] (\operatorname {LegendreP}(n-1,K[1])-K[1] \operatorname {LegendreP}(n,K[1]))}{\left (K[1]^2-1\right )^2 \left (n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},K[1]^2\right ) \left ((n+1) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},\frac {n+3}{2},\frac {5}{2},K[1]^2\right ) K[1]^2+3 \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[1]^2\right )\right )-3 (n+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {n}{2},\frac {1}{2},K[1]^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[1]^2\right )\right )}dK[1]+i x \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},x^2\right ) \int _1^x\frac {3 i n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},K[2]^2\right ) (\operatorname {LegendreP}(n-1,K[2])-K[2] \operatorname {LegendreP}(n,K[2]))}{\left (K[2]^2-1\right )^2 \left (n \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},K[2]^2\right ) \left ((n+1) \operatorname {Hypergeometric2F1}\left (1-\frac {n}{2},\frac {n+3}{2},\frac {5}{2},K[2]^2\right ) K[2]^2+3 \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[2]^2\right )\right )-3 (n+1) \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {n}{2},\frac {n}{2},\frac {1}{2},K[2]^2\right ) \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},K[2]^2\right )\right )}dK[2]+c_1 \operatorname {Hypergeometric2F1}\left (-\frac {n}{2}-\frac {1}{2},\frac {n}{2},\frac {1}{2},x^2\right )+i c_2 x \operatorname {Hypergeometric2F1}\left (-\frac {n}{2},\frac {n+1}{2},\frac {3}{2},x^2\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
n = symbols("n")
y = Function("y")
ode = Eq(-n*(n + 1)*y(x) + (x**2 - 1)*Derivative(y(x), (x, 2)) + Diff(LegendreP(n, x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : solve: Cannot solve -n*(n + 1)*y(x) + (x**2 - 1)*Derivative(y(x), (x, 2)) + Diff(LegendreP(n, x), x)