55.31.4 problem 152

Internal problem ID [13925]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-5
Problem number : 152
Date solved : Thursday, October 02, 2025 at 08:09:09 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 33

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

\[ y = c_1 \left (x +\sqrt {x^{2}-1}\right )^{-n}+c_2 \left (x +\sqrt {x^{2}-1}\right )^{n} \]

✓ Mathematica. Time used: 0.026 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to c_1 \cosh \left (n \log \left (\sqrt {x^2-1}+x\right )\right )+i c_2 \sinh \left (n \log \left (\sqrt {x^2-1}+x\right )\right ) \end{align*}

✗ Sympy

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

False