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 = 0.235022 (sec), leaf count = 80
\[\left \{\left \{y(x)\to -\frac {\sqrt {c_1} \sqrt {W\left (\frac {2 e^{\frac {2 \left (c_2+x\right )}{c_1}}}{c_1}\right )}}{\sqrt {2}}\right \},\left \{y(x)\to \frac {\sqrt {c_1} \sqrt {W\left (\frac {2 e^{\frac {2 \left (c_2+x\right )}{c_1}}}{c_1}\right )}}{\sqrt {2}}\right \}\right \}\]
Maple ✓
cpu = 0.051 (sec), leaf count = 23
\[ \left \{ \left ( y \left ( x \right ) \right ) ^{2}+{\it \_C1}\,\ln \left ( y \left ( x \right ) \right ) -x-{\it \_C2}=0,y \left ( x \right ) ={\it \_C1} \right \} \] 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),'implicit')
Maple raw output
y(x) = _C1, y(x)^2+_C1*ln(y(x))-x-_C2 = 0