Internal problem ID [9913]
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.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{\prime } y-y-\frac {A}{x}+\frac {A^{2}}{x^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 117
dsolve(y(x)*diff(y(x),x)-y(x)=A*x^(-1)-A^2*x^(-3),y(x), singsol=all)
\[ y \relax (x ) = \frac {\left (c_{1} x^{2}-A \,{\mathrm e}^{\RootOf \left (2 \textit {\_Z} A \,{\mathrm e}^{2 \textit {\_Z}}-x^{2} {\mathrm e}^{2 \textit {\_Z}}+2 c_{1} x^{2} {\mathrm e}^{\textit {\_Z}}-c_{1}^{2} x^{2}-2 A \,{\mathrm e}^{2 \textit {\_Z}}+2 A c_{1} {\mathrm e}^{\textit {\_Z}}\right )}\right ) {\mathrm e}^{-\RootOf \left (2 \textit {\_Z} A \,{\mathrm e}^{2 \textit {\_Z}}-x^{2} {\mathrm e}^{2 \textit {\_Z}}+2 c_{1} x^{2} {\mathrm e}^{\textit {\_Z}}-c_{1}^{2} x^{2}-2 A \,{\mathrm e}^{2 \textit {\_Z}}+2 A c_{1} {\mathrm e}^{\textit {\_Z}}\right )}}{x} \]
✓ Solution by Mathematica
Time used: 0.351 (sec). Leaf size: 63
DSolve[y[x]*y'[x]-y[x]==A*x^(-1)-A^2*x^(-3),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [x^2 \left (-\frac {1}{A}+\frac {2 x^2 \log \left (\frac {x^2}{A+x y(x)}\right )+2 A-c_1 x^2+2 x y(x)}{\left (A-x^2+x y(x)\right )^2}\right )=0,y(x)\right ] \]