Internal problem ID [9922]
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: 16.
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}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 57
dsolve(y(x)*diff(y(x),x)-y(x)=A*1/x,y(x), singsol=all)
\[ c_{1}+\frac {-\erf \left (\frac {\left (x -y \relax (x )\right ) \sqrt {2}}{2 \sqrt {-A}}\right ) \sqrt {2}\, \sqrt {\pi }\, x -2 \,{\mathrm e}^{\frac {\left (x -y \relax (x )\right )^{2}}{2 A}} \sqrt {-A}}{x} = 0 \]
✓ Solution by Mathematica
Time used: 0.493 (sec). Leaf size: 64
DSolve[y[x]*y'[x]-y[x]==A*1/x,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [-\frac {x}{\sqrt {A}}=\frac {2 e^{\frac {(x-y(x))^2}{2 A}}}{\sqrt {2 \pi } \text {Erfi}\left (\frac {y(x)-x}{\sqrt {2} \sqrt {A}}\right )+2 c_1},y(x)\right ] \]