2.32.3 Problem 184

Internal problem ID [13845]
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 : 184
Date solved : Friday, December 19, 2025 at 03:22:10 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

2.32.3.1 ✓ Maple. Time used: 0.019 (sec). Leaf size: 95

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

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

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 successful 
   <- special function solution successful 
      -> Trying to convert hypergeometric functions to elementary form... 
      <- elementary form is not straightforward to achieve - returning special \ 
function solution free of uncomputed integrals 
   <- Kovacics algorithm successful
 

2.32.3.2 ✓ Mathematica. Time used: 1.234 (sec). Leaf size: 62

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

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

2.32.3.3 ✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-b*y(x) - x**3*Derivative(y(x), (x, 2)))/(x*(a*x + b)) cannot be solved by the factorable group method