2.32.19 Problem 201

Internal problem ID [13861]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 2, Second-Order Differential Equations. section 2.1.2-6
Problem number : 201
Date solved : Friday, December 19, 2025 at 04:51:41 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.32.19.1 ✓ Maple. Time used: 0.044 (sec). Leaf size: 79

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

\[ y = {\mathrm e}^{-a x} \left (a x +2\right ) \left (c_2 \int {\mathrm e}^{a \int \frac {a^{2} x^{4}+2 a \,x^{3}+\left (a^{2} b +2\right ) x^{2}+4 a b x +2 b}{\left (a \,x^{3}+x^{2}+b \right ) \left (a x +2\right )}d x}d x +c_1 \right ) \]

Maple trace

Methods for second order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
checking if the LODE has constant coefficients 
checking if the LODE is of Euler type 
trying a symmetry of the form [xi=0, eta=F(x)] 
checking if the LODE is missing y 
-> Trying a Liouvillian solution using Kovacics algorithm 
   A Liouvillian solution exists 
   Reducible group (found an exponential solution) 
   Group is reducible, not completely reducible 
   Solution has integrals. Trying a special function solution free of integrals\ 
... 
   -> Trying a solution in terms of special functions: 
      -> Bessel 
      -> elliptic 
      -> Legendre 
      -> Kummer 
         -> hyper3: Equivalence to 1F1 under a power @ Moebius 
      -> hypergeometric 
         -> heuristic approach 
         -> hyper3: Equivalence to 2F1, 1F1 or 0F1 under a power @ Moebius 
      -> Mathieu 
         -> Equivalence to the rational form of Mathieu ODE under a power @ Mo\ 
ebius 
      -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a \ 
power @ Moebius 
   No special function solution was found. 
<- Kovacics algorithm successful
 

2.32.19.2 ✗ Mathematica

ode=(a*x^3+x^2+b)*D[y[x],{x,2}]+a^2*x*(x^2-b)*D[y[x],x]-a^3*b*x*y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Timed out

2.32.19.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
y = Function("y") 
ode = Eq(-a**3*b*x*y(x) + a**2*x*(-b + x**2)*Derivative(y(x), x) + (a*x**3 + b + x**2)*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

False
 

Python version: 3.12.3 (main, Aug 14 2025, 17:47:21) [GCC 13.3.0] 
Sympy version 1.14.0
 

classify_ode(ode,func=y(x)) 
 
('factorable', '2nd_power_series_ordinary')