55.30.31 problem 140

Internal problem ID [13913]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-4
Problem number : 140
Date solved : Thursday, October 02, 2025 at 08:08:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.093 (sec). Leaf size: 199

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

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

✓ Mathematica. Time used: 1.708 (sec). Leaf size: 67

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

\begin{align*} y(x)&\to \frac {e^{-a x} (a x+1) \left (c_2 \int _1^x\frac {a^2 e^{\frac {b}{K[1]}+a K[1]} K[1]^{1-a b}}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \end{align*}

✗ Sympy

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

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