4.24.8 $y^{\prime }(x)+y(x)\log (y^{\prime }(x))-xy(x)-y(x)\log (y(x))=0$

ODE
$$y^{\prime }(x)+y(x)\log (y^{\prime }(x))-xy(x)-y(x)\log (y(x))=0$$ ODE Classification

[_separable]

Book solution method
No Missing Variables ODE, Solve for $x$

Mathematica ✓
cpu = 0.0092828 (sec), leaf count = 24

$$\left\{\{y(x)\to c_1{e}^{\frac{1}{2}W\left(e^x\right)(W\left(e^x\right)+2)}\}\right\}$$

Maple ✓
cpu = 0.153 (sec), leaf count = 17

$$\{y\left(x\right)=\mathit{\_}\mathit{C}\mathit{1}{\mathrm{e}}^{\frac{\mathit{l}\mathit{a}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{t}\mathit{W}\left({\mathrm{e}}^x\right)(\mathit{l}\mathit{a}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathit{t}\mathit{W}\left({\mathrm{e}}^x\right)+2)}{2}}\}$$ 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))