##### 4.14.13 $$x \left (x^2-6 y(x)^2\right ) y'(x)=4 y(x) \left (x^2+3 y(x)^2\right )$$

ODE
$x \left (x^2-6 y(x)^2\right ) y'(x)=4 y(x) \left (x^2+3 y(x)^2\right )$ ODE Classiﬁcation

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.382495 (sec), leaf count = 67

$\left \{\left \{y(x)\to \frac {e^{c_1}-\sqrt {-24 x^6+e^{2 c_1}}}{12 x^2}\right \},\left \{y(x)\to \frac {\sqrt {-24 x^6+e^{2 c_1}}+e^{c_1}}{12 x^2}\right \}\right \}$

Maple
cpu = 0.254 (sec), leaf count = 47

$\left [y \left (x \right ) = -\frac {\textit {\_C1} \left (-1+\sqrt {-\frac {24 x^{6}}{\textit {\_C1}^{2}}+1}\right )}{12 x^{2}}, y \left (x \right ) = \frac {\textit {\_C1} \left (1+\sqrt {-\frac {24 x^{6}}{\textit {\_C1}^{2}}+1}\right )}{12 x^{2}}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^C[1] - Sqrt[E^(2*C[1]) - 24*x^6])/(12*x^2)}, {y[x] -> (E^C[1] + Sqr
t[E^(2*C[1]) - 24*x^6])/(12*x^2)}}

Maple raw input

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

Maple raw output

[y(x) = -1/12*_C1/x^2*(-1+(-24/_C1^2*x^6+1)^(1/2)), y(x) = 1/12*_C1/x^2*(1+(-24/
_C1^2*x^6+1)^(1/2))]