61.24.39 problem 39

Internal problem ID [12373]
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 : 39
Date solved : Monday, March 31, 2025 at 05:22:51 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 92

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

\[ c_1 +\frac {{\mathrm e}^{\frac {a x +y}{2 a}} \sqrt {\frac {-a x +a -y}{a x +y}}\, y}{\sqrt {\frac {a}{a x +y}}\, \left (a x +y\right ) x}+\int _{}^{\frac {a}{a x +y}}\frac {{\mathrm e}^{\frac {1}{2 \textit {\_a}}} \sqrt {\textit {\_a} -1}}{\sqrt {\textit {\_a}}}d \textit {\_a} = 0 \]

✗ Mathematica

ode=y[x]*D[y[x],x]+a*(x-2)*x^(-1)*y[x]==2*a^2*(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") 
y = Function("y") 
ode = Eq(-2*a**2*(x - 1)/x + a*(x - 2)*y(x)/x + y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out