55.24.62 problem 62

Internal problem ID [13691]
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 : 62
Date solved : Friday, September 05, 2025 at 11:04:22 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

\begin{align*} y y^{\prime }-a \left (\left (n +2 k -3\right ) x +3-2 k \right ) x^{-k} y&=a^{2} \left (\left (n +k -1\right ) x^{2}-\left (n +2 k -3\right ) x +k -2\right ) x^{1-2 k} \end{align*}

✗ Maple

ode:=y(x)*diff(y(x),x)-a*((n+2*k-3)*x+3-2*k)*x^(-k)*y(x) = a^2*((n+k-1)*x^2-(n+2*k-3)*x+k-2)*x^(1-2*k); 
dsolve(ode,y(x), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=y[x]*D[y[x],x]-a*((n+2*k-3)*x+3-2*k)*x^(-k)*y[x]==a^2*((n+k-1)*x^2-(n+2*k-3)*x+k-2)*x^(1-2*k); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

Not solved

✗ Sympy

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

Timed Out