Internal problem ID [9981]
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 1.3.2. Equations of the
form \(y y'=f(x) y+1\)
Problem number: 3.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, [_Abel, `2nd type`, `class B`]]
\[ \boxed {y^{\prime } y-\left (a -\frac {1}{a x}\right ) y-1=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 39
dsolve(y(x)*diff(y(x),x)=(a-1/(a*x))*y(x)+1,y(x), singsol=all)
\[ y \left (x \right ) = \frac {a^{2} x -\operatorname {RootOf}\left (-{\mathrm e}^{\textit {\_Z}}-\operatorname {Ei}_{1}\left (-\textit {\_Z} \right ) a^{2} x +c_{1} a^{2} x \right )}{a} \]
✓ Solution by Mathematica
Time used: 0.168 (sec). Leaf size: 37
DSolve[y[x]*y'[x]==(a-1/(a*x))*y[x]+1,y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\operatorname {ExpIntegralEi}(a (a x-y(x)))+c_1=\frac {e^{a (a x-y(x))}}{a^2 x},y(x)\right ] \]