55.24.15 problem 15

Internal problem ID [13644]
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 : 15
Date solved : Sunday, October 12, 2025 at 04:20:26 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 279

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

\[ -\frac {27 \left (m -1\right ) \left (-54 a \left (m +2\right ) \left (m +\frac {1}{2}\right ) m^{4} x \int _{}^{\frac {9 m \left (y \left (m -1\right )+3 a \left (\frac {1}{3}+\left (x -\frac {1}{3}\right ) m \right )\right )}{\left (m -1\right ) \left (2 m +1\right ) \left (m +2\right ) \left (-y+a \right )}}\frac {\textit {\_a} {\left (\left (m^{2}+m -2\right ) \textit {\_a} -9 m \right )}^{\frac {1}{1+m}} {\left (\left (2 m^{2}-m -1\right ) \textit {\_a} +9 m \right )}^{\frac {m}{1+m}}}{8 \left (\left (m^{2}-\frac {1}{2} m -\frac {1}{2}\right ) \textit {\_a} +\frac {9 m}{2}\right ) \left (\left (m^{2}+m -2\right ) \textit {\_a} -9 m \right ) {\left (\left (m^{2}+\frac {5}{2} m +1\right ) \textit {\_a} +\frac {9 m}{2}\right )}^{2}}d \textit {\_a} +\left (\frac {1}{3}+\left (x -\frac {1}{3}\right ) m \right ) \left (-y+a \right ) \left (\frac {m^{2} \left (\left (x -1\right ) a +y\right )}{\left (2 m +1\right ) \left (-y+a \right )}\right )^{\frac {1}{1+m}} \left (\frac {\left (a m x +a -y\right ) m}{\left (m +2\right ) \left (-y+a \right )}\right )^{\frac {m}{1+m}}-\frac {2 a c_1 \left (m +2\right ) \left (m +\frac {1}{2}\right ) m x}{27}\right )}{m \left (2 m^{3}+3 m^{2}-3 m -2\right ) a x} = 0 \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out