Internal problem ID [10063]
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`]]
\[ \boxed {y^{\prime } y-a \left (1+2 n +2 n \left (1+n \right ) x \right ) {\mathrm e}^{\left (1+n \right ) x} y+a^{2} n \left (1+n \right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (1+n \right ) x}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 130
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)
\[ y \left (x \right ) = \frac {\left (1+2 n^{2} x +\left (\sqrt {-\frac {\left (1+n \right )^{2}}{n^{2}}}\, \tan \left (\frac {\operatorname {RootOf}\left (2 x \,n^{2} {\mathrm e}^{\textit {\_a} +\textit {\_Z}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (1+n \right )^{2}}{n^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {\left (1+n \right )^{2}}{n^{2}}}\, n +2 n x \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+n \,{\mathrm e}^{\textit {\_a} +\textit {\_Z}}+2 c_{1} n \,{\mathrm e}^{\textit {\_a}}+{\mathrm e}^{\textit {\_a} +\textit {\_Z}}\right ) \sqrt {-\frac {\left (1+n \right )^{2}}{n^{2}}}}{2}\right )+2 x +1\right ) n \right ) a \,{\mathrm e}^{\left (1+n \right ) x}}{2+2 n} \]
✗ 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