Internal problem ID [10753]
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: 6.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y y^{\prime }+a \left (1-\frac {1}{x}\right ) y=a^{2}} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 28
dsolve(y(x)*diff(y(x),x)+a*(1-x^(-1))*y(x)=a^2,y(x), singsol=all)
\[ y \left (x \right ) = -a x +\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}}-\operatorname {Ei}_{1}\left (-\textit {\_Z} \right ) x +x c_{1} \right ) a \]
✓ Solution by Mathematica
Time used: 0.208 (sec). Leaf size: 30
DSolve[y[x]*y'[x]+a*(1-x^(-1))*y[x]==a^2,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\operatorname {ExpIntegralEi}\left (x+\frac {y(x)}{a}\right )+c_1=\frac {e^{\frac {y(x)}{a}+x}}{x},y(x)\right ] \]