61.24.59 problem 59

Internal problem ID [12472]
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 : Tuesday, January 28, 2025 at 07:59:53 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

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

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

\[ y = \frac {2 \left (-\frac {\sqrt {-n^{2}}\, x \tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x -2 a n \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+n x \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+2 c_{1} x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )}{2}+\left (a -\frac {x}{2}\right ) n \right ) x^{-n}}{\tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x -2 a n \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+n x \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+2 c_{1} x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}+n} \]

✗ Solution by Mathematica

Time used: 0.000 (sec). Leaf size: 0

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

Not solved