54.3.225 problem 1241

Internal problem ID [12520]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1241
Date solved : Friday, October 03, 2025 at 03:19:45 AM
CAS classification : [_Gegenbauer]

\begin{align*} \left (x^{2}-1\right ) y^{\prime \prime }-2 x y^{\prime }-\left (v +2\right ) \left (v -1\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.017 (sec). Leaf size: 23

ode:=(x^2-1)*diff(diff(y(x),x),x)-2*x*diff(y(x),x)-(v+2)*(v-1)*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_2 \operatorname {LegendreQ}\left (v , 2, x\right )+c_1 \operatorname {LegendreP}\left (v , 2, x\right )\right ) \left (x^{2}-1\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 26

ode=(1 - v)*(2 + v)*y[x] - 2*x*D[y[x],x] + (-1 + x^2)*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \left (x^2-1\right ) (c_1 P_v^2(x)+c_2 Q_v^2(x)) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
v = symbols("v") 
y = Function("y") 
ode = Eq(-2*x*Derivative(y(x), x) - (v - 1)*(v + 2)*y(x) + (x**2 - 1)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False