3 Mathematica. Time used: 0.003 (sec). Leaf size: 12

ode=x*y[x]*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]

\begin{align*} y(x)&\to 0\\ y(x)&\to c_1 \end{align*}