##### 4.13.46 $$x \left (2 x^2+y(x)^2\right ) y'(x)=y(x) \left (2 x^2+3 y(x)^2\right )$$

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

[[_homogeneous, class A], _rational, _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.346947 (sec), leaf count = 56

$\left \{\left \{y(x)\to -\frac {\sqrt {2} x}{\sqrt {W\left (\frac {2 e^{-2 c_1}}{x^4}\right )}}\right \},\left \{y(x)\to \frac {\sqrt {2} x}{\sqrt {W\left (\frac {2 e^{-2 c_1}}{x^4}\right )}}\right \}\right \}$

Maple
cpu = 0.147 (sec), leaf count = 25

$\left [y \left (x \right ) = {\mathrm e}^{\frac {\LambertW \left (\frac {2 \,{\mathrm e}^{-4 \textit {\_C1}}}{x^{4}}\right )}{2}+2 \textit {\_C1}} x^{3}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> -((Sqrt[2]*x)/Sqrt[ProductLog[2/(E^(2*C[1])*x^4)]])}, {y[x] -> (Sqrt[2
]*x)/Sqrt[ProductLog[2/(E^(2*C[1])*x^4)]]}}

Maple raw input

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

Maple raw output

[y(x) = exp(1/2*LambertW(2/exp(_C1)^4/x^4)+2*_C1)*x^3]