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 = 0.319259 (sec), leaf count = 34
\[\text {Solve}\left [\log \left (W\left (4 x e^{2 y(x)}\right )+2\right )+2 y(x)+c_1=W\left (4 x e^{2 y(x)}\right ),y(x)\right ]\]
Maple ✓
cpu = 0.048 (sec), leaf count = 67
\[\left [y \left (x \right ) = \frac {\ln \left (\frac {-1+\sqrt {16 \textit {\_C1} x +1}}{4 x}\right )}{2}-\frac {1}{2}+\frac {\sqrt {16 \textit {\_C1} x +1}}{2}, y \left (x \right ) = \frac {\ln \left (-\frac {1+\sqrt {16 \textit {\_C1} x +1}}{4 x}\right )}{2}-\frac {1}{2}-\frac {\sqrt {16 \textit {\_C1} x +1}}{2}\right ]\] 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))
Maple raw output
[y(x) = 1/2*ln(1/4/x*(-1+(16*_C1*x+1)^(1/2)))-1/2+1/2*(16*_C1*x+1)^(1/2), y(x) = 1/2*ln(-1/4*(1+(16*_C1*x+1)^(1/2))/x)-1/2-1/2*(16*_C1*x+1)^(1/2)]