61.24.59 problem 59

Internal problem ID [12393]
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 : 59
Date solved : Monday, March 31, 2025 at 05:28:37 AM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 151

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

\[ y = \frac {2 x^{-n} \left (-\frac {\sqrt {-n^{2}}\, x \tan \left (\frac {\operatorname {RootOf}\left (-2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} \sqrt {-n^{2}}\, x +2 c_1 x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )}{2}+\left (a -\frac {x}{2}\right ) n \right )}{\tan \left (\frac {\operatorname {RootOf}\left (-2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} \sqrt {-n^{2}}\, x +2 c_1 x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}+n} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

Timed Out