2.32.6 Problem 187

Internal problem ID [13848]
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 : 187
Date solved : Friday, December 19, 2025 at 03:28:19 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.32.6.1 ✓ Maple. Time used: 0.033 (sec). Leaf size: 49

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

\[ y = {\mathrm e}^{-a x} \left (a x +1\right ) \left (c_2 \int \frac {x \,{\mathrm e}^{\frac {2 a \,x^{3}+2 a b x +b}{2 x^{2}}}}{\left (a x +1\right )^{2}}d x +c_1 \right ) \]

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
 

2.32.6.2 ✓ Mathematica. Time used: 0.498 (sec). Leaf size: 70

ode=x^3*D[y[x],{x,2}]+(a*x^3-x^2+a*b*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^{a K[1]+\frac {2 a K[1] b+b}{2 K[1]^2}} K[1]}{(a K[1]+1)^2}dK[1]+c_1\right )}{a} \end{align*}

2.32.6.3 ✗ 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**3*Derivative(y(x), (x, 2)) + (a*b*x + a*x**3 + b - x**2)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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

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',)