23.3.552 problem 559

Internal problem ID [6266]
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 : 559
Date solved : Friday, October 03, 2025 at 01:58:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.146 (sec). Leaf size: 101

ode:=b*y(x)+a*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);
 

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

✗ Mathematica

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

False