Internal problem ID [10775]
Internal file name [OUTPUT/9723_Monday_June_06_2022_04_47_51_PM_99545307/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: 39.
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 B`]]
Unable to solve or complete the solution.
\[ \boxed {y y^{\prime }+\frac {a \left (x -2\right ) y}{x}=\frac {2 a^{2} \left (x -1\right )}{x}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } x +a x y-2 a^{2} x -2 a y+2 a^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {-a x y+2 a^{2} x +2 a y-2 a^{2}}{x y} \end {array} \]
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 found: 2 potential symmetries. Proceeding with integration step <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 116
dsolve(y(x)*diff(y(x),x)+a*(x-2)*x^(-1)*y(x)=2*a^2*(x-1)*x^(-1),y(x), singsol=all)
\[ \frac {\sqrt {\frac {\left (1-x \right ) a -y \left (x \right )}{a x +y \left (x \right )}}\, {\mathrm e}^{\frac {a x +y \left (x \right )}{2 a}} y \left (x \right )+x \left (\int _{}^{\frac {a}{a x +y \left (x \right )}}\frac {\sqrt {\textit {\_a} -1}\, {\mathrm e}^{\frac {1}{2 \textit {\_a}}}}{\sqrt {\textit {\_a}}}d \textit {\_a} +c_{1} \right ) \sqrt {\frac {a}{a x +y \left (x \right )}}\, \left (a x +y \left (x \right )\right )}{\sqrt {\frac {a}{a x +y \left (x \right )}}\, x \left (a x +y \left (x \right )\right )} = 0 \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y[x]*y'[x]+a*(x-2)*x^(-1)*y[x]==2*a^2*(x-1)*x^(-1),y[x],x,IncludeSingularSolutions -> True]
Not solved