Internal problem ID [7835]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 255.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {x \left (y x -3\right ) y^{\prime }+x y^{2}-y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.288 (sec). Leaf size: 74
dsolve(x*(x*y(x)-3)*diff(y(x),x)+x*y(x)^2-y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = -\frac {3 \LambertW \left (\frac {2 \left (-\frac {x^{2}}{8}\right )^{\frac {1}{3}} c_{1}}{3}\right )}{x} \\ y \relax (x ) = -\frac {3 \LambertW \left (\frac {\left (-\frac {x^{2}}{8}\right )^{\frac {1}{3}} c_{1} \left (-1+i \sqrt {3}\right )}{3}\right )}{x} \\ y \relax (x ) = -\frac {3 \LambertW \left (-\frac {\left (-\frac {x^{2}}{8}\right )^{\frac {1}{3}} c_{1} \left (1+i \sqrt {3}\right )}{3}\right )}{x} \\ \end{align*}
✓ Solution by Mathematica
Time used: 30.004 (sec). Leaf size: 35
DSolve[x*(x*y[x]-3)*y'[x]+x*y[x]^2-y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {3 \operatorname {ProductLog}\left (e^{-1+\frac {9 c_1}{2^{2/3}}} x^{2/3}\right )}{x} \\ y(x)\to 0 \\ \end{align*}