2.24.10 Problem 15

Internal problem ID [13574]
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 : Friday, December 19, 2025 at 07:16:38 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*}

Unknown ode type.

2.24.10.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 279

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

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

Maple trace

Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
trying Bernoulli 
trying separable 
trying inverse linear 
trying homogeneous types: 
trying Chini 
differential order: 1; looking for linear symmetries 
trying exact 
trying Abel 
Looking for potential symmetries 
Looking for potential symmetries 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-\frac {a \left (\left (m -1\right ) x +1\right ) y \left (x \right )}{x}=\frac {a^{2} \left (m x +1\right ) \left (x -1\right )}{x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {\frac {a \left (\left (m -1\right ) x +1\right ) y \left (x \right )}{x}+\frac {a^{2} \left (m x +1\right ) \left (x -1\right )}{x}}{y \left (x \right )} \end {array} \]

2.24.10.2 ✗ 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

2.24.10.3 ✗ 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