54.3.354 problem 1371

Internal problem ID [12649]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1371
Date solved : Friday, October 03, 2025 at 03:45:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }&=-\frac {2 x y^{\prime }}{x^{2}-1}-\frac {\left (-a^{2}-\lambda \left (x^{2}-1\right )\right ) y}{\left (x^{2}-1\right )^{2}} \end{align*}

✓ Maple. Time used: 0.019 (sec). Leaf size: 37

ode:=diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2-lambda*(x^2-1))/(x^2-1)^2*y(x); 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.025 (sec). Leaf size: 48

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

\begin{align*} y(x)&\to c_1 P_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x) \end{align*}

✗ Sympy

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

False