54.3.289 problem 1306

Internal problem ID [12584]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 2, linear second order
Problem number : 1306
Date solved : Friday, October 03, 2025 at 03:38:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 69

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

\[ y = \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {x +1}{x -1}\right ) \left (c_1 +c_2 \int \frac {1}{x \operatorname {HeunD}\left (0, 8 a +4 b , 0, 8 a -4 b , \frac {x +1}{x -1}\right )^{2}}d x \right ) \]

✗ Mathematica

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

NotImplementedError : The given ODE a*y(x) + a*y(x)/x**2 + b*y(x)/x + x*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be solved by the factorable group method