61.24.8 problem 8

Internal problem ID [12421]
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 : 8
Date solved : Tuesday, January 28, 2025 at 07:58:28 PM
CAS classification : [_rational, [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 153

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)
 

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

✗ Solution by Mathematica

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

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

Not solved