61.24.41 problem 41

Internal problem ID [12375]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.3. Abel Equations of the Second Kind. subsection 1.3.3-2.
Problem number : 41
Date solved : Monday, March 31, 2025 at 05:22:57 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

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