4.14.42 $$\left (a y(x)^3+x^3\right ) y'(x)=x^2 y(x)$$

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

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.370482 (sec), leaf count = 108

$\left \{\left \{y(x)\to \frac {x}{\sqrt [3]{a} \sqrt [3]{W\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}}\right \},\left \{y(x)\to -\frac {\sqrt [3]{-1} x}{\sqrt [3]{a} \sqrt [3]{W\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}}\right \},\left \{y(x)\to \frac {(-1)^{2/3} x}{\sqrt [3]{a} \sqrt [3]{W\left (\frac {x^3 e^{-\frac {3 c_1}{a}}}{a}\right )}}\right \}\right \}$

Maple
cpu = 0.175 (sec), leaf count = 23

$\left [y \left (x \right ) = \left (\frac {1}{a \LambertW \left (\frac {\textit {\_C1} \,x^{3}}{a}\right )}\right )^{\frac {1}{3}} x\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> x/(a^(1/3)*ProductLog[x^3/(a*E^((3*C[1])/a))]^(1/3))}, {y[x] -> -(((-1
)^(1/3)*x)/(a^(1/3)*ProductLog[x^3/(a*E^((3*C[1])/a))]^(1/3)))}, {y[x] -> ((-1)^
(2/3)*x)/(a^(1/3)*ProductLog[x^3/(a*E^((3*C[1])/a))]^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = (1/a/LambertW(1/a*_C1*x^3))^(1/3)*x]