##### 4.14.47 $$x \left (2 x^3-y(x)^3\right ) y'(x)=y(x) \left (x^3-2 y(x)^3\right )$$

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

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.642971 (sec), leaf count = 433

$\left \{\left \{y(x)\to \frac {1}{3} \left (e^{c_1} x^2+\frac {\sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}{\sqrt [3]{2}}+\frac {\sqrt [3]{2} e^{2 c_1} x^4}{\sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right )\right \},\left \{y(x)\to \frac {e^{c_1} x^2}{3}+\frac {i \left (\sqrt {3}+i\right ) \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}{6 \sqrt [3]{2}}-\frac {i \left (\sqrt {3}-i\right ) e^{2 c_1} x^4}{3\ 2^{2/3} \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right \},\left \{y(x)\to \frac {e^{c_1} x^2}{3}-\frac {i \left (\sqrt {3}-i\right ) \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}{6 \sqrt [3]{2}}+\frac {i \left (\sqrt {3}+i\right ) e^{2 c_1} x^4}{3\ 2^{2/3} \sqrt [3]{2 e^{3 c_1} x^6-27 x^3+3 \sqrt {81 x^6-12 e^{3 c_1} x^9}}}\right \}\right \}$

Maple
cpu = 0.089 (sec), leaf count = 355

$\left [y \left (x \right ) = \left (\frac {\left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{6}+\frac {2 \textit {\_C1}^{2} x^{2}}{3 \left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {x \textit {\_C1}}{3}\right ) x, y \left (x \right ) = \left (-\frac {\left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{12}-\frac {\textit {\_C1}^{2} x^{2}}{3 \left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {x \textit {\_C1}}{3}-\frac {i \sqrt {3}\, \left (\frac {\left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 \textit {\_C1}^{2} x^{2}}{3 \left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x, y \left (x \right ) = \left (-\frac {\left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{12}-\frac {\textit {\_C1}^{2} x^{2}}{3 \left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}+\frac {x \textit {\_C1}}{3}+\frac {i \sqrt {3}\, \left (\frac {\left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}{6}-\frac {2 \textit {\_C1}^{2} x^{2}}{3 \left (-108+8 x^{3} \textit {\_C1}^{3}+12 \sqrt {-12 x^{3} \textit {\_C1}^{3}+81}\right )^{\frac {1}{3}}}\right )}{2}\right ) x\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (E^C[1]*x^2 + (2^(1/3)*E^(2*C[1])*x^4)/(-27*x^3 + 2*E^(3*C[1])*x^6 + 3
*Sqrt[81*x^6 - 12*E^(3*C[1])*x^9])^(1/3) + (-27*x^3 + 2*E^(3*C[1])*x^6 + 3*Sqrt[
81*x^6 - 12*E^(3*C[1])*x^9])^(1/3)/2^(1/3))/3}, {y[x] -> (E^C[1]*x^2)/3 - ((I/3)
*(-I + Sqrt[3])*E^(2*C[1])*x^4)/(2^(2/3)*(-27*x^3 + 2*E^(3*C[1])*x^6 + 3*Sqrt[81
*x^6 - 12*E^(3*C[1])*x^9])^(1/3)) + ((I/6)*(I + Sqrt[3])*(-27*x^3 + 2*E^(3*C[1])
*x^6 + 3*Sqrt[81*x^6 - 12*E^(3*C[1])*x^9])^(1/3))/2^(1/3)}, {y[x] -> (E^C[1]*x^2
)/3 + ((I/3)*(I + Sqrt[3])*E^(2*C[1])*x^4)/(2^(2/3)*(-27*x^3 + 2*E^(3*C[1])*x^6
+ 3*Sqrt[81*x^6 - 12*E^(3*C[1])*x^9])^(1/3)) - ((I/6)*(-I + Sqrt[3])*(-27*x^3 +
2*E^(3*C[1])*x^6 + 3*Sqrt[81*x^6 - 12*E^(3*C[1])*x^9])^(1/3))/2^(1/3)}}

Maple raw input

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

Maple raw output

[y(x) = (1/6*(-108+8*x^3*_C1^3+12*(-12*_C1^3*x^3+81)^(1/2))^(1/3)+2/3*_C1^2*x^2/
(-108+8*x^3*_C1^3+12*(-12*_C1^3*x^3+81)^(1/2))^(1/3)+1/3*x*_C1)*x, y(x) = (-1/12
*(-108+8*x^3*_C1^3+12*(-12*_C1^3*x^3+81)^(1/2))^(1/3)-1/3*_C1^2*x^2/(-108+8*x^3*
_C1^3+12*(-12*_C1^3*x^3+81)^(1/2))^(1/3)+1/3*x*_C1-1/2*I*3^(1/2)*(1/6*(-108+8*x^
3*_C1^3+12*(-12*_C1^3*x^3+81)^(1/2))^(1/3)-2/3*_C1^2*x^2/(-108+8*x^3*_C1^3+12*(-
12*_C1^3*x^3+81)^(1/2))^(1/3)))*x, y(x) = (-1/12*(-108+8*x^3*_C1^3+12*(-12*_C1^3
*x^3+81)^(1/2))^(1/3)-1/3*_C1^2*x^2/(-108+8*x^3*_C1^3+12*(-12*_C1^3*x^3+81)^(1/2
))^(1/3)+1/3*x*_C1+1/2*I*3^(1/2)*(1/6*(-108+8*x^3*_C1^3+12*(-12*_C1^3*x^3+81)^(1
/2))^(1/3)-2/3*_C1^2*x^2/(-108+8*x^3*_C1^3+12*(-12*_C1^3*x^3+81)^(1/2))^(1/3)))*
x]