Internal problem ID [9925]
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: 19.
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-2 x -\frac {A}{x^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 169
dsolve(y(x)*diff(y(x),x)-y(x)=2*x+A*x^(-2),y(x), singsol=all)
\[ c_{1}+\frac {-6 \sqrt {3}\, \arctanh \left (\frac {\sqrt {\frac {x \left (A^{2}\right )^{\frac {1}{3}}}{A}}\, \left (2 x -y \relax (x )\right )}{\sqrt {\frac {\left (4 x^{3}-4 x^{2} y \relax (x )+y \relax (x )^{2} x +2 A \right ) \left (A^{2}\right )^{\frac {1}{3}}}{y \relax (x )^{2} A}}\, y \relax (x )}\right ) A x \sqrt {\frac {x \left (A^{2}\right )^{\frac {1}{3}}}{A}}+\sqrt {\frac {\left (4 x^{3}-4 x^{2} y \relax (x )+y \relax (x )^{2} x +2 A \right ) \left (A^{2}\right )^{\frac {1}{3}}}{y \relax (x )^{2} A}}\, y \relax (x ) \left (-2 x^{3}-x^{2} y \relax (x )+y \relax (x )^{2} x +2 A \right ) \sqrt {3}}{x \sqrt {\frac {x \left (A^{2}\right )^{\frac {1}{3}}}{A}}} = 0 \]
✓ Solution by Mathematica
Time used: 1.23 (sec). Leaf size: 233
DSolve[y[x]*y'[x]-y[x]==2*x+A*x^(-2),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [c_1=-\frac {i \sqrt {-\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}} \left (-6 \sqrt {A} x^{3/2} \sinh ^{-1}\left (\frac {\sqrt {x} (2 x-y(x))}{\sqrt {2} \sqrt {A}}\right )+x^2 (-y(x)) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+x y(x)^2 \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}+2 \left (A-x^3\right ) \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}\right )}{4 \sqrt {A} x^{3/2} \sqrt {\frac {2 A+4 x^3-4 x^2 y(x)+x y(x)^2}{A}}},y(x)\right ] \]