##### 4.14.12 $$x \left (6 y(x)^2+x\right ) y'(x)-3 y(x)^3+x y(x)=0$$

ODE
$x \left (6 y(x)^2+x\right ) y'(x)-3 y(x)^3+x y(x)=0$ ODE Classiﬁcation

[[_homogeneous, class G], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.372945 (sec), leaf count = 64

$\left \{\left \{y(x)\to -\frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\right \}\right \}$

Maple
cpu = 0.247 (sec), leaf count = 25

$\left [y \left (x \right ) = \frac {{\mathrm e}^{-\frac {\LambertW \left (\frac {6 \,{\mathrm e}^{3 \textit {\_C1}}}{x^{3}}\right )}{2}+\frac {3 \textit {\_C1}}{2}}}{x}\right ]$ Mathematica raw input

DSolve[x*y[x] - 3*y[x]^3 + x*(x + 6*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((Sqrt[x]*Sqrt[ProductLog[(6*E^(3*C[1]))/x^3]])/Sqrt[6])}, {y[x] -> (
Sqrt[x]*Sqrt[ProductLog[(6*E^(3*C[1]))/x^3]])/Sqrt[6]}}

Maple raw input

dsolve(x*(x+6*y(x)^2)*diff(y(x),x)+x*y(x)-3*y(x)^3 = 0, y(x))

Maple raw output

[y(x) = 1/x*exp(-1/2*LambertW(6*exp(3*_C1)/x^3)+3/2*_C1)]