60.3.216 problem 1231
Internal
problem
ID
[11212]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
2,
linear
second
order
Problem
number
:
1231
Date
solved
:
Sunday, March 30, 2025 at 07:46:17 PM
CAS
classification
:
[_Gegenbauer]
\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-v \left (v +1\right ) y&=0 \end{align*}
✓ Maple. Time used: 0.024 (sec). Leaf size: 52
ode:=(x^2-1)*diff(diff(y(x),x),x)-v*(v+1)*y(x) = 0;
dsolve(ode,y(x), singsol=all);
\[
y = -\left (x -1\right ) \left (x +1\right ) \left (\operatorname {hypergeom}\left (\left [1-\frac {v}{2}, \frac {3}{2}+\frac {v}{2}\right ], \left [\frac {3}{2}\right ], x^{2}\right ) c_2 x +c_1 \operatorname {hypergeom}\left (\left [1+\frac {v}{2}, \frac {1}{2}-\frac {v}{2}\right ], \left [\frac {1}{2}\right ], x^{2}\right )\right )
\]
✓ Mathematica. Time used: 0.101 (sec). Leaf size: 56
ode=-(v*(1 + v)*y[x]) + (-1 + x^2)*D[y[x],{x,2}] == 0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to c_1 \operatorname {Hypergeometric2F1}\left (-\frac {v}{2}-\frac {1}{2},\frac {v}{2},\frac {1}{2},x^2\right )+i c_2 x \operatorname {Hypergeometric2F1}\left (-\frac {v}{2},\frac {v+1}{2},\frac {3}{2},x^2\right )
\]
✓ Sympy. Time used: 0.349 (sec). Leaf size: 94
from sympy import *
x = symbols("x")
v = symbols("v")
y = Function("y")
ode = Eq(-v*(v + 1)*y(x) + (x**2 - 1)*Derivative(y(x), (x, 2)),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {\left (C_{1} \sqrt {x^{2}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4} - \frac {\sqrt {\left (2 v + 1\right )^{2}}}{4}, \frac {\sqrt {\left (2 v + 1\right )^{2}}}{4} + \frac {1}{4} \\ \frac {3}{2} \end {matrix}\middle | {x^{2}} \right )} + C_{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {\sqrt {\left (2 v + 1\right )^{2}}}{4} - \frac {1}{4}, \frac {\sqrt {\left (2 v + 1\right )^{2}}}{4} - \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {x^{2}} \right )}\right ) \sqrt [4]{x^{2}}}{\sqrt {x}}
\]