61.31.20 problem 201

Internal problem ID [12622]
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 : Monday, March 31, 2025 at 06:06:32 AM
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*}

✓ Maple. Time used: 0.075 (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);
 

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

✗ 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

✗ 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