Internal problem ID [10049]
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`]]
\[ \boxed {y^{\prime } y-\left (\left (-1+2 n \right ) x -a n \right ) x^{-n -1} y-n \left (-a +x \right ) x^{-2 n}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 150
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)
\[ y \left (x \right ) = -\frac {\left (\sqrt {-n^{2}}\, x \tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x -2 \,{\mathrm e}^{\textit {\_Z}} n a \,{\mathrm e}^{\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z}} {\mathrm e}^{\textit {\_a}}+2 x c_{1} {\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )-2 a n +n x \right ) x^{-n}}{\sqrt {-n^{2}}\, \tan \left (\frac {\operatorname {RootOf}\left (-\sqrt {-n^{2}}\, \tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} x -2 \,{\mathrm e}^{\textit {\_Z}} n a \,{\mathrm e}^{\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z}} {\mathrm e}^{\textit {\_a}}+2 x c_{1} {\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )+n} \]
✗ 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