2.22.10 Problem 13

Internal problem ID [13505]
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 1.3.1-2. Solvable equations and their solutions
Problem number : 13
Date solved : Friday, December 19, 2025 at 05:12:06 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

Unknown ode type.

2.22.10.1 ✓ Maple. Time used: 0.002 (sec). Leaf size: 166

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

\[ \frac {\left (\frac {-2 m x y-2 m A -x^{2}}{2 y x +2 A}\right )^{\frac {1}{m +1}} y 2^{-\frac {m}{m +1}} \left (y x +A \right ) \left (\frac {\left (-2 m -1\right ) x^{2}+2 m x y+2 m A}{y x +A}\right )^{\frac {2 m +1}{m +1}}-x \left (A \int _{}^{-\frac {x^{2}}{2 y x +2 A}}\frac {\left (-m +\textit {\_a} \right )^{\frac {1}{m +1}} \left (\left (2 \textit {\_a} +1\right ) m +\textit {\_a} \right )^{\frac {2 m +1}{m +1}}}{\textit {\_a}^{2}}d \textit {\_a} -c_1 \right )}{x} = 0 \]

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 
found: 2 potential symmetries. Proceeding with integration step 
<- Abel successful
 

Maple step by step

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \left (x \right ) \left (\frac {d}{d x}y \left (x \right )\right )-y \left (x \right )=\frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \frac {d}{d x}y \left (x \right ) \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & \frac {d}{d x}y \left (x \right )=\frac {y \left (x \right )+\frac {\left (2 m +1\right ) x}{4 m^{2}}+\frac {A}{x}-\frac {A^{2}}{x^{3}}}{y \left (x \right )} \end {array} \]

2.22.10.2 ✗ Mathematica

ode=y[x]*D[y[x],x]-y[x]==(2*m+1)/(4*m^2)*x+A*1/x-A^2*1/(x^3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

2.22.10.3 ✗ Sympy

from sympy import * 
x = symbols("x") 
A = symbols("A") 
m = symbols("m") 
y = Function("y") 
ode = Eq(A**2/x**3 - A/x + y(x)*Derivative(y(x), x) - y(x) - x*(2*m + 1)/(4*m**2),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out