Internal problem ID [10001]
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: 7.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y y^{\prime }-a \left (1-\frac {b}{x}\right ) y-b \,a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.002 (sec). Leaf size: 30
dsolve(y(x)*diff(y(x),x)-a*(1-b*x^(-1))*y(x)=a^2*b,y(x), singsol=all)
\[ y \relax (x ) = -\RootOf \left (-\expIntegral \left (1, -\textit {\_Z} \right ) x -{\mathrm e}^{\textit {\_Z}} b +c_{1} x \right ) a b +x a \]
✓ Solution by Mathematica
Time used: 0.181 (sec). Leaf size: 45
DSolve[y[x]*y'[x]-a*(1-b*x^(-1))*y[x]==a^2*b,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\text {Ei}\left (\frac {a x-y(x)}{a b}\right )+c_1=\frac {b e^{\frac {a x-y(x)}{a b}}}{x},y(x)\right ] \]