Internal problem ID [10067]
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: 73.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_Abel, 2nd type, class A]]
Solve \begin {gather*} \boxed {y^{\prime } y-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y+a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 160
dsolve(y(x)*diff(y(x),x)-a*(1+2*n+2*n*(n+1)*x)*exp((n+1)*x)*y(x)=-a^2*n*(n+1)*(1+n*x)*x*exp(2*(n+1)*x),y(x), singsol=all)
\[ y \relax (x ) = \frac {a \left (\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n \tan \left (\frac {\RootOf \left (2 \,{\mathrm e}^{\textit {\_Z}} n^{2} x +\left (\int _{}^{-\frac {\textit {\_Z} \sqrt {-\frac {n^{2}+2 n +1}{n^{2}}}}{\sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}}\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) {\mathrm e}^{-\textit {\_a}}d \textit {\_a} \right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n +2 \,{\mathrm e}^{\textit {\_Z}} n x +{\mathrm e}^{\textit {\_Z}} n +2 c_{1} n +{\mathrm e}^{\textit {\_Z}}\right ) \sqrt {-\frac {n^{2}+2 n +1}{n^{2}}}}{2}\right )+2 n^{2} x +2 x n +n +1\right ) {\mathrm e}^{\left (n +1\right ) x}}{2 n +2} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-a*(1+2*n+2*n*(n+1)*x)*Exp[(n+1)*x]*y[x]==-a^2*n*(n+1)*(1+n*x)*x*Exp[2*(n+1)*x],y[x],x,IncludeSingularSolutions -> True]
Not solved