##### 4.12.40 $$3 x^4+8 x^3 y(x) y'(x)-6 x^2 y(x)^2-y(x)^4=0$$

ODE
$3 x^4+8 x^3 y(x) y'(x)-6 x^2 y(x)^2-y(x)^4=0$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.299104 (sec), leaf count = 78

$\left \{\left \{y(x)\to -\frac {\sqrt {-x^2 \left (3+e^{8 c_1} x\right )}}{\sqrt {-1+e^{8 c_1} x}}\right \},\left \{y(x)\to \frac {\sqrt {-x^2 \left (3+e^{8 c_1} x\right )}}{\sqrt {-1+e^{8 c_1} x}}\right \}\right \}$

Maple
cpu = 0.136 (sec), leaf count = 54

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

DSolve[3*x^4 - 6*x^2*y[x]^2 - y[x]^4 + 8*x^3*y[x]*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -(Sqrt[-(x^2*(3 + E^(8*C[1])*x))]/Sqrt[-1 + E^(8*C[1])*x])}, {y[x] ->
Sqrt[-(x^2*(3 + E^(8*C[1])*x))]/Sqrt[-1 + E^(8*C[1])*x]}}

Maple raw input

dsolve(8*x^3*y(x)*diff(y(x),x)+3*x^4-6*x^2*y(x)^2-y(x)^4 = 0, y(x))

Maple raw output

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