Internal problem ID [10002]
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: 8.
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-x^{n -1} \left (\left (2 n +1\right ) x +a n \right ) y+n \,x^{2 n} \left (a +x \right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 166
dsolve(y(x)*diff(y(x),x)=x^(n-1)*((1+2*n)*x+a*n)*y(x)-n*x^(2*n)*(x+a),y(x), singsol=all)
\[ y \relax (x ) = \frac {\tan \left (\frac {\RootOf \left (-2 \,{\mathrm e}^{\textit {\_Z}} n a -{\mathrm e}^{\textit {\_Z}} n x +\left (\int _{}^{-\textit {\_Z}}\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) \sqrt {-n^{2}}\, x +2 c_{1} x \right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}\, x^{n +1}-2 a n \,x^{n}-x^{n +1} n}{\tan \left (\frac {\RootOf \left (-2 \,{\mathrm e}^{\textit {\_Z}} n a -{\mathrm e}^{\textit {\_Z}} n x +\left (\int _{}^{-\textit {\_Z}}\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) \sqrt {-n^{2}}\, x +2 c_{1} x \right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}-n} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]==x^(n-1)*((1+2*n)*x+a*n)*y[x]-n*x^(2*n)*(x+a),y[x],x,IncludeSingularSolutions -> True]
Not solved