23.3.535 problem 541

Internal problem ID [6249]
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 : 541
Date solved : Friday, October 03, 2025 at 01:57:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (c \,x^{4}+b \,x^{2}+a \right ) y+x^{3} y^{\prime }+x^{4} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.061 (sec). Leaf size: 81

ode:=(c*x^4+b*x^2+a)*y(x)+x^3*diff(y(x),x)+x^4*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \operatorname {HeunD}\left (0, c +b +a , -2 a +2 c , a -b +c , \frac {x^{2}+1}{x^{2}-1}\right ) \left (c_1 +c_2 \int \frac {1}{x \operatorname {HeunD}\left (0, c +b +a , -2 a +2 c , a -b +c , \frac {x^{2}+1}{x^{2}-1}\right )^{2}}d x \right ) \]

✗ Mathematica

ode=(a + b*x^2 + c*x^4)*y[x] + x^3*D[y[x],x] + x^4*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") 
c = symbols("c") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 2)) + x**3*Derivative(y(x), x) + (a + b*x**2 + c*x**4)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE a*y(x)/x**3 + b*y(x)/x + c*x*y(x) + x*Derivative(y(x), (x, 2)) +