2.5.1.4 ✓ Mathematica. Time used: 9.868 (sec). Leaf size: 50

ode=x*D[y[x],x]^2-y[x]+2*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{gather*} \text {Solve}\left [\left \{x=\frac {2 \log (K[1])-2 K[1]}{(K[1]-1)^2}+\frac {c_1}{(K[1]-1)^2},y(x)=x K[1]^2+2 K[1]\right \},\{y(x),K[1]\}\right ] \end{gather*}