54.3.178 problem 1192

Internal problem ID [12473]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1192
Date solved : Wednesday, October 01, 2025 at 01:45:33 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.052 (sec). Leaf size: 51

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

\[ y = \left ({\mathrm e}^{-\frac {1}{x}} \operatorname {HeunD}\left (-4, 3, -8, 5, \frac {x -1}{x +1}\right ) c_2 +{\mathrm e}^{-x} \operatorname {HeunD}\left (4, 3, -8, 5, \frac {x -1}{x +1}\right ) c_1 \right ) \sqrt {x} \]

✓ Mathematica. Time used: 0.172 (sec). Leaf size: 35

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

\begin{align*} y(x)&\to e^{-x} \left (c_2 \int _1^xe^{K[1]-\frac {1}{K[1]}}dK[1]+c_1\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-x**2*Derivative(y(x), (x, 2)) + y(x))/(x**2 - 1) cannot be solved by the factorable group method