ODE
\[ 4 x y'(x)+\log \left (y'(x)\right )-2 y(x)=0 \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
Clairaut’s equation and related types, d’Alembert’s equation (also call Lagrange’s)
Mathematica ✓
cpu = 0.114256 (sec), leaf count = 34
\[\text {Solve}\left [c_1+\log \left (W\left (4 x e^{2 y(x)}\right )+2\right )+2 y(x)=W\left (4 x e^{2 y(x)}\right ),y(x)\right ]\]
Maple ✓
cpu = 0.008 (sec), leaf count = 33
\[ \left \{ [x \left ( {\it \_T} \right ) ={\frac {1}{{{\it \_T}}^{2}} \left ( -{\frac {{\it \_T}}{2}}+{\it \_C1} \right ) },y \left ( {\it \_T} \right ) ={\frac {\ln \left ( {\it \_T} \right ) }{2}}+{\frac {-{\it \_T}+2\,{\it \_C1}}{{\it \_T}}}] \right \} \] Mathematica raw input
DSolve[Log[y'[x]] - 2*y[x] + 4*x*y'[x] == 0,y[x],x]
Mathematica raw output
Solve[C[1] + Log[2 + ProductLog[4*E^(2*y[x])*x]] + 2*y[x] == ProductLog[4*E^(2*y[x])*x], y[x]]
Maple raw input
dsolve(ln(diff(y(x),x))+4*x*diff(y(x),x)-2*y(x) = 0, y(x),'implicit')
Maple raw output
[x(_T) = 1/_T^2*(-1/2*_T+_C1), y(_T) = 1/2*ln(_T)+(-_T+2*_C1)/_T]