ODE
\[ a+x y'(x)+\log \left (y'(x)\right )=0 \] ODE Classification
[_quadrature]
Book solution method
Missing Variables ODE, Dependent variable missing, Solve for \(x\)
Mathematica ✓
cpu = 0.00995125 (sec), leaf count = 30
\[\left \{\left \{y(x)\to \frac {1}{2} W\left (e^{-a} x\right )^2+W\left (e^{-a} x\right )+c_1\right \}\right \}\]
Maple ✓
cpu = 0.056 (sec), leaf count = 24
\[ \left \{ y \left ( x \right ) ={\frac {1}{2} \left ( {\it lambertW} \left ( {\frac {x}{{{\rm e}^{a}}}} \right ) \right ) ^{2}}+{\it lambertW} \left ( {\frac {x}{{{\rm e}^{a}}}} \right ) +{\it \_C1} \right \} \] Mathematica raw input
DSolve[a + Log[y'[x]] + x*y'[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> C[1] + ProductLog[x/E^a] + ProductLog[x/E^a]^2/2}}
Maple raw input
dsolve(ln(diff(y(x),x))+x*diff(y(x),x)+a = 0, y(x),'implicit')
Maple raw output
y(x) = 1/2*LambertW(x/exp(a))^2+LambertW(x/exp(a))+_C1