Internal problem ID [10053]
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. Equations of
the form \(y y'=f_1(x) y+f_0(x)\)
Problem number: 59.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {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}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 156
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)
\[ y \relax (x ) = -\frac {\left (\sqrt {-n^{2}}\, x \tan \left (\frac {\RootOf \left (-2 \,{\mathrm e}^{\textit {\_Z}} n a +{\mathrm e}^{\textit {\_Z}} n x +\left (\int _{}^{-\textit {\_Z}}\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) \sqrt {-n^{2}}\, x +2 x c_{1}\right ) \sqrt {-n^{2}}}{2}\right )-2 a n +x n \right ) x^{-n}}{\sqrt {-n^{2}}\, \tan \left (\frac {\RootOf \left (-2 \,{\mathrm e}^{\textit {\_Z}} n a +{\mathrm e}^{\textit {\_Z}} n x +\left (\int _{}^{-\textit {\_Z}}\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) \sqrt {-n^{2}}\, x +2 x c_{1}\right ) \sqrt {-n^{2}}}{2}\right )+n} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[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