4.11.19 \(x^2 e^{-\frac {2 y(x)}{x}}+x y(x) y'(x)-y(x)^2=0\)

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 = 0.0454222 (sec), leaf count = 25

\[\left \{\left \{y(x)\to \frac {1}{2} x \left (W\left (\frac {4 \left (c_1-\log (x)\right )}{e}\right )+1\right )\right \}\right \}\]

Maple
cpu = 0.012 (sec), leaf count = 32

\[ \left \{ {\frac {x-2\,y \relax (x ) }{4\,x} \left ({{\rm e}^{-2\,{\frac {y \relax (x ) }{x}}}} \right ) ^{-1}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

1/4*(x-2*y(x))/x/exp(-2*y(x)/x)-ln(x)-_C1 = 0