55.24.20 problem 20

Internal problem ID [13649]
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 : 20
Date solved : Wednesday, October 01, 2025 at 10:47:53 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y y^{\prime }-\frac {a \left (1+\frac {2 b}{x^{2}}\right ) y}{2}&=\frac {a^{2} \left (3 x +\frac {4 b}{x}\right )}{16} \end{align*}

✗ Maple

ode:=y(x)*diff(y(x),x)-1/2*a*(1+2*b/x^2)*y(x) = 1/16*a^2*(3*x+4*b/x); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*D[y[x],x]-1/2*a*(1+2*b*x^(-2))*y[x]==1/16*a^2*(3*x+4*b/x); 
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*(4*b/x + 3*x)/16 - a*(2*b/x**2 + 1)*y(x)/2 + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -a**2*b/(4*x*y(x)) - 3*a**2*x/(16*y(x)) - a*b/x**2 - a/2 + Derivative(y(x), x) cannot be solved by the factorable group method