Internal problem ID [10795]
Internal file name [OUTPUT/9743_Monday_June_06_2022_09_31_13_PM_94900719/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: 59.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[_rational, [_Abel, `2nd type`, `class A`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }-\left (\left (2 n -1\right ) x -n a \right ) x^{-n -1} y=n \left (x -a \right ) x^{-2 n}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime }-\left (\left (2 n -1\right ) x -n a \right ) x^{-n -1} y=n \left (x -a \right ) x^{-2 n} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\left (\left (2 n -1\right ) x -n a \right ) x^{-n -1} y+n \left (x -a \right ) x^{-2 n}}{y} \end {array} \]
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: 0.0 (sec). Leaf size: 151
dsolve(y(x)*diff(y(x),x)-((2*n-1)*x-a*n)*x^(-n-1)*y(x)=n*(x-a)*x^(-2*n),y(x), singsol=all)
\[ y \left (x \right ) = \frac {2 \left (-\frac {\sqrt {-n^{2}}\, x \tan \left (\frac {\operatorname {RootOf}\left (-2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} \sqrt {-n^{2}}\, x +2 c_{1} x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right )}{2}+\left (a -\frac {x}{2}\right ) n \right ) x^{-n}}{\tan \left (\frac {\operatorname {RootOf}\left (-2 a n \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}+n x \,{\mathrm e}^{\textit {\_Z} +\textit {\_a}}-\tan \left (\frac {\textit {\_a} \sqrt {-n^{2}}}{2}\right ) \textit {\_Z} \sqrt {-n^{2}}\, x +2 c_{1} x \,{\mathrm e}^{\textit {\_a}}\right ) \sqrt {-n^{2}}}{2}\right ) \sqrt {-n^{2}}+n} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]-((2*n-1)*x-a*n)*x^(-n-1)*y[x]==n*(x-a)*x^(-2*n),y[x],x,IncludeSingularSolutions -> True]
Not solved