55.24.56 problem 56

Internal problem ID [13685]
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 : 56
Date solved : Sunday, October 12, 2025 at 04:32:32 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class B`]]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 143

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

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

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out