ODE
\[ x^2 e^{-\frac {2 y(x)}{x}}+x y(x) y'(x)-y(x)^2=0 \] ODE Classification
[[_homogeneous, `class A`], _dAlembert]
Book solution method
Homogeneous equation
Mathematica ✓
cpu = 0.407711 (sec), leaf count = 25
\[\left \{\left \{y(x)\to \frac {1}{2} x \left (1+W\left (\frac {4 (-\log (x)+c_1)}{e}\right )\right )\right \}\right \}\]
Maple ✓
cpu = 0.036 (sec), leaf count = 18
\[\left [y \left (x \right ) = \frac {\left (\LambertW \left (-4 \left (\ln \left (x \right )+\textit {\_C1} \right ) {\mathrm e}^{-1}\right )+1\right ) x}{2}\right ]\] Mathematica raw input
DSolve[x^2/E^((2*y[x])/x) - y[x]^2 + x*y[x]*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> (x*(1 + ProductLog[(4*(C[1] - Log[x]))/E]))/2}}
Maple raw input
dsolve(x*y(x)*diff(y(x),x)+x^2*exp(-2*y(x)/x)-y(x)^2 = 0, y(x))
Maple raw output
[y(x) = 1/2*(LambertW(-4*(ln(x)+_C1)*exp(-1))+1)*x]