54.3.223 problem 1239

Internal problem ID [12518]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1239
Date solved : Friday, October 03, 2025 at 03:19:43 AM
CAS classification : [_Gegenbauer]

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

✓ Maple. Time used: 0.018 (sec). Leaf size: 35

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

\[ y = c_1 \operatorname {LegendreP}\left (\frac {\sqrt {1+4 l}}{2}-\frac {1}{2}, x\right )+c_2 \operatorname {LegendreQ}\left (\frac {\sqrt {1+4 l}}{2}-\frac {1}{2}, x\right ) \]

✓ Mathematica. Time used: 0.021 (sec). Leaf size: 46

ode=-(l*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 c_1 \operatorname {LegendreP}\left (\frac {1}{2} \left (\sqrt {4 l+1}-1\right ),x\right )+c_2 \operatorname {LegendreQ}\left (\frac {1}{2} \left (\sqrt {4 l+1}-1\right ),x\right ) \end{align*}

✗ Sympy

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

False