61.22.13 problem 13

Internal problem ID [12259]
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 : Monday, March 31, 2025 at 04:56:44 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.009 (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);
 

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

✗ 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

✗ 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