23.3.323 problem 325

Internal problem ID [6037]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 3. THE DIFFERENTIAL EQUATION IS LINEAR AND OF SECOND ORDER, page 311
Problem number : 325
Date solved : Tuesday, September 30, 2025 at 02:20:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.038 (sec). Leaf size: 51

ode:=-y(x)-(-x^2+1)*diff(y(x),x)+x^2*diff(diff(y(x),x),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: 0.175 (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)) - (1 - x**2)*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