Internal problem ID [10809]
Internal file name [OUTPUT/9757_Thursday_June_09_2022_12_57_31_AM_11538751/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y=-a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y=-a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {a \left (1+2 n +2 n \left (n +1\right ) x \right ) {\mathrm e}^{\left (n +1\right ) x} y-a^{2} n \left (n +1\right ) \left (n x +1\right ) x \,{\mathrm e}^{2 \left (n +1\right ) x}}{y} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact trying Abel Looking for potential symmetries Looking for potential symmetries <- Abel successful`
✓ Solution by Maple
Time used: 0.016 (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)
\[ y \left (x \right ) = \frac {{\mathrm e}^{\left (n +1\right ) x} a \left (1+2 x \,n^{2}+\left (\tan \left (\frac {\operatorname {RootOf}\left (2 x \,n^{2} {\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \textit {\_Z} \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}\, n +2 n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+2 c_{1} n \,{\mathrm e}^{\textit {\_a}}+{\mathrm e}^{\textit {\_Z} +\textit {\_a}}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}}{2}\right ) \sqrt {-\frac {\left (n +1\right )^{2}}{n^{2}}}+2 x +1\right ) n \right )}{2 n +2} \]
✗ 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