ODE
\[ y(x) y''(x)+4 y(x) y'(x)^3-y'(x)^2=0 \] ODE Classification
[[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_y_y1]]
Book solution method
TO DO
Mathematica ✓
cpu = 0.473278 (sec), leaf count = 80
\[\left \{\left \{y(x)\to -\frac {\sqrt {c_1} \sqrt {W\left (\frac {2 e^{\frac {2 (x+c_2)}{c_1}}}{c_1}\right )}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {c_1} \sqrt {W\left (\frac {2 e^{\frac {2 (x+c_2)}{c_1}}}{c_1}\right )}}{\sqrt {2}}\right \}\right \}\]
Maple ✓
cpu = 1.692 (sec), leaf count = 45
\[\left [y \left (x \right ) = \textit {\_C1}, y \left (x \right ) = {\mathrm e}^{-\frac {\textit {\_C1} \LambertW \left (\frac {2 \,{\mathrm e}^{\frac {2 x}{\textit {\_C1}}} {\mathrm e}^{\frac {2 \textit {\_C2}}{\textit {\_C1}}}}{\textit {\_C1}}\right )-2 \textit {\_C2} -2 x}{2 \textit {\_C1}}}\right ]\] Mathematica raw input
DSolve[-y'[x]^2 + 4*y[x]*y'[x]^3 + y[x]*y''[x] == 0,y[x],x]
Mathematica raw output
{{y[x] -> -((Sqrt[C[1]]*Sqrt[ProductLog[(2*E^((2*(x + C[2]))/C[1]))/C[1]]])/Sqrt[2])}, {y[x] -> (Sqrt[C[1]]*Sqrt[ProductLog[(2*E^((2*(x + C[2]))/C[1]))/C[1]]])/Sqrt[2]}}
Maple raw input
dsolve(y(x)*diff(diff(y(x),x),x)-diff(y(x),x)^2+4*y(x)*diff(y(x),x)^3 = 0, y(x))
Maple raw output
[y(x) = _C1, y(x) = exp(-1/2*(_C1*LambertW(2/_C1*exp(x/_C1)^2*exp(1/_C1*_C2)^2)-2*_C2-2*x)/_C1)]