23.3.547 problem 553

Internal problem ID [6261]
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 : 553
Date solved : Friday, October 03, 2025 at 01:57:55 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.134 (sec). Leaf size: 54

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

\[ y = c_1 \operatorname {HeunC}\left (2, -\frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}+\frac {a^{2}}{4}, \frac {1}{x^{2}+1}\right )+\frac {c_2 \operatorname {HeunC}\left (2, \frac {1}{2}, -\frac {1}{2}, -1, \frac {7}{8}+\frac {a^{2}}{4}, \frac {1}{x^{2}+1}\right )}{\sqrt {x^{2}+1}} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

False