Internal problem ID [10655]
Internal file name [OUTPUT/9603_Monday_June_06_2022_03_13_21_PM_1762640/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. Form \(y y'-y=f(x)\). subsection 1.3.1-2.
Solvable equations and their solutions
Problem number: 7.
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.
\[ \boxed {y y^{\prime }-y=\frac {A}{x}-\frac {A^{2}}{x^{3}}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y y^{\prime } x^{3}-y x^{3}-A \,x^{2}+A^{2}=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {y x^{3}+A \,x^{2}-A^{2}}{y x^{3}} \end {array} \]
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 <- Abel successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 118
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-1)-A^2*x^(-3),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-c_{1} x^{2}+A \,{\mathrm e}^{\operatorname {RootOf}\left (2 \textit {\_Z} A \,{\mathrm e}^{2 \textit {\_Z}}-x^{2} {\mathrm e}^{2 \textit {\_Z}}+2 c_{1} x^{2} {\mathrm e}^{\textit {\_Z}}-c_{1}^{2} x^{2}-2 A \,{\mathrm e}^{2 \textit {\_Z}}+2 A c_{1} {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\operatorname {RootOf}\left (2 \textit {\_Z} A \,{\mathrm e}^{2 \textit {\_Z}}-x^{2} {\mathrm e}^{2 \textit {\_Z}}+2 c_{1} x^{2} {\mathrm e}^{\textit {\_Z}}-c_{1}^{2} x^{2}-2 A \,{\mathrm e}^{2 \textit {\_Z}}+2 A c_{1} {\mathrm e}^{\textit {\_Z}}\right )}}{x} \]
✓ Solution by Mathematica
Time used: 0.569 (sec). Leaf size: 63
DSolve[y[x]*y'[x]-y[x]==A*x^(-1)-A^2*x^(-3),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [x^2 \left (-\frac {1}{A}+\frac {2 x^2 \log \left (\frac {x^2}{A+x y(x)}\right )+2 A-c_1 x^2+2 x y(x)}{\left (A-x^2+x y(x)\right )^2}\right )=0,y(x)\right ] \]