23.3.554 problem 561

Internal problem ID [6268]
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 : 561
Date solved : Friday, October 03, 2025 at 01:58:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.109 (sec). Leaf size: 225

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

\[ y = \left (a^{2}+x^{2}\right )^{-\frac {a^{2}+\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}-1}{4 a^{2}}} {\mathrm e}^{\frac {1}{2 a^{2}+2 x^{2}}} x^{\frac {a^{2}+\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}-1}{2 a^{2}}} \left (\sqrt {a^{2}+x^{2}}\, \operatorname {HeunC}\left (\frac {1}{2 a^{2}}, -\frac {1}{2}, \frac {\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}}{2 a^{2}}, \frac {4 a^{2}+1}{8 a^{4}}, \frac {a^{2}-6}{8 a^{2}}, \frac {a^{2}}{a^{2}+x^{2}}\right ) c_1 +\operatorname {HeunC}\left (\frac {1}{2 a^{2}}, \frac {1}{2}, \frac {\sqrt {a^{4}-2 a^{2}-4 b^{2}+1}}{2 a^{2}}, \frac {4 a^{2}+1}{8 a^{4}}, \frac {a^{2}-6}{8 a^{2}}, \frac {a^{2}}{a^{2}+x^{2}}\right ) c_2 \right ) \]

✗ Mathematica

ode=b^2*y[x] + x*(a^2 + 2*x^2)*D[y[x],x] + x^2*(a^2 + 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(b**2*y(x) + x**2*(a**2 + x**2)**2*Derivative(y(x), (x, 2)) + x*(a**2 + 2*x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

ValueError : Expected Expr or iterable but got None