61.24.73 problem 73

Internal problem ID [12486]
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 : 73
Date solved : Tuesday, January 28, 2025 at 08:01:39 PM
CAS classification : [[_Abel, `2nd type`, `class A`]]

\begin{align*} 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} \end{align*}

✓ Solution by Maple

Time used: 0.006 (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)
 

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

✗ Solution by Mathematica

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

DSolve[y[x]*D[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],x,IncludeSingularSolutions -> True]
 

Not solved