ODE
\[ y'(x)+y(x) \log \left (y'(x)\right )-x y(x)-y(x) \log (y(x))=0 \] ODE Classification
[_separable]
Book solution method
No Missing Variables ODE, Solve for \(x\)
Mathematica ✓
cpu = 0.0092828 (sec), leaf count = 24
\[\left \{\left \{y(x)\to c_1 e^{\frac {1}{2} W\left (e^x\right ) \left (W\left (e^x\right )+2\right )}\right \}\right \}\]
Maple ✓
cpu = 0.153 (sec), leaf count = 17
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{{\rm e}^{{\frac {{\it lambertW} \left ( {{\rm e}^{x}} \right ) \left ( {\it lambertW} \left ( {{\rm e}^{x}} \right ) +2 \right ) }{2}}}} \right \} \] Mathematica raw input
DSolve[-(x*y[x]) - Log[y[x]]*y[x] + Log[y'[x]]*y[x] + y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> E^((ProductLog[E^x]*(2 + ProductLog[E^x]))/2)*C[1]}}
Maple raw input
dsolve(y(x)*ln(diff(y(x),x))+diff(y(x),x)-y(x)*ln(y(x))-x*y(x) = 0, y(x),'implicit')
Maple raw output
y(x) = _C1*exp(1/2*LambertW(exp(x))*(LambertW(exp(x))+2))