ODE
\[ x y''(x)+2 x y'(x)^2-2 y'(x)=0 \] ODE Classification
[[_2nd_order, _missing_y], _Liouville, [_2nd_order, _reducible, _mu_xy]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.017444 (sec), leaf count = 23
\[\left \{\left \{y(x)\to \frac {1}{2} \log \left (3 c_1+2 x^3\right )+c_2\right \}\right \}\]
Maple ✓
cpu = 0.214 (sec), leaf count = 19
\[ \left \{ {\it \_C1}\,{x}^{3}-{\it \_C2}+{\frac {{{\rm e}^{2\,y \left ( x \right ) }}}{2}}=0 \right \} \] Mathematica raw input
DSolve[-2*y'[x] + 2*x*y'[x]^2 + x*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[2] + Log[2*x^3 + 3*C[1]]/2}}
Maple raw input
dsolve(x*diff(diff(y(x),x),x)+2*x*diff(y(x),x)^2-2*diff(y(x),x) = 0, y(x),'implicit')
Maple raw output
_C1*x^3-_C2+1/2*exp(2*y(x)) = 0