##### 4.12.32 $$x^2 (x-2 y(x)) y'(x)=2 x^3-4 x y(x)^2+y(x)^3$$

ODE
$x^2 (x-2 y(x)) y'(x)=2 x^3-4 x y(x)^2+y(x)^3$ ODE Classiﬁcation

[[_homogeneous, class A], _rational, [_Abel, 2nd type, class C], _dAlembert]

Book solution method
Homogeneous equation

Mathematica
cpu = 0.482076 (sec), leaf count = 101

$\left \{\left \{y(x)\to \frac {2 x^3-\sqrt {e^{2 c_1} x^2 \left (-3 x^2+e^{2 c_1}\right )}}{x^2+e^{2 c_1}}\right \},\left \{y(x)\to \frac {2 x^3+\sqrt {e^{2 c_1} x^2 \left (-3 x^2+e^{2 c_1}\right )}}{x^2+e^{2 c_1}}\right \}\right \}$

Maple
cpu = 0.268 (sec), leaf count = 75

$\left [y \left (x \right ) = \frac {\left (3 x^{2} \textit {\_C1} -\sqrt {3 x^{2} \textit {\_C1} +1}-1\right ) x}{x^{2} \textit {\_C1} -1}-x, y \left (x \right ) = \frac {\left (3 x^{2} \textit {\_C1} +\sqrt {3 x^{2} \textit {\_C1} +1}-1\right ) x}{x^{2} \textit {\_C1} -1}-x\right ]$ Mathematica raw input

DSolve[x^2*(x - 2*y[x])*y'[x] == 2*x^3 - 4*x*y[x]^2 + y[x]^3,y[x],x]

Mathematica raw output

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

Maple raw input

dsolve(x^2*(x-2*y(x))*diff(y(x),x) = 2*x^3-4*x*y(x)^2+y(x)^3, y(x))

Maple raw output

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