##### 4.14.49 $$x \left (x^3-2 y(x)^3\right ) y'(x)=y(x) \left (2 x^3-y(x)^3\right )$$

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

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.606561 (sec), leaf count = 331

$\left \{\left \{y(x)\to \frac {\sqrt [3]{2} \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}+2 \sqrt [3]{3} e^{c_1} x}{6^{2/3} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}}\right \},\left \{y(x)\to \frac {i \sqrt [3]{2} \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}+3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}}\right \},\left \{y(x)\to \frac {\sqrt [3]{2} \sqrt [6]{3} \left (-1-i \sqrt {3}\right ) \left (-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}\right ){}^{2/3}-2 \left (\sqrt {3}-3 i\right ) e^{c_1} x}{2\ 2^{2/3} 3^{5/6} \sqrt [3]{-9 x^3+\sqrt {81 x^6-12 e^{3 c_1} x^3}}}\right \}\right \}$

Maple
cpu = 0.075 (sec), leaf count = 447

$\left [y \left (x \right ) = \frac {12^{\frac {1}{3}} \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{6 \textit {\_C1}}+\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}, y \left (x \right ) = -\frac {12^{\frac {1}{3}} \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{12 \textit {\_C1}}-\frac {x 12^{\frac {2}{3}}}{12 \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {12^{\frac {1}{3}} \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}\right )}{2}, y \left (x \right ) = -\frac {12^{\frac {1}{3}} \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{12 \textit {\_C1}}-\frac {x 12^{\frac {2}{3}}}{12 \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {12^{\frac {1}{3}} \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}{6 \textit {\_C1}}-\frac {x 12^{\frac {2}{3}}}{6 \left (x \left (-9 x^{2} \textit {\_C1} +\sqrt {3}\, \sqrt {\frac {x \left (27 x^{3} \textit {\_C1}^{3}-4\right )}{\textit {\_C1}}}\right ) \textit {\_C1}^{2}\right )^{\frac {1}{3}}}\right )}{2}\right ]$ Mathematica raw input

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

Mathematica raw output

{{y[x] -> (2*3^(1/3)*E^C[1]*x + 2^(1/3)*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x
^6])^(2/3))/(6^(2/3)*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x^6])^(1/3))}, {y[x]
 -> (-2*(3*I + Sqrt[3])*E^C[1]*x + I*2^(1/3)*3^(1/6)*(I + Sqrt[3])*(-9*x^3 + Sqr
t[-12*E^(3*C[1])*x^3 + 81*x^6])^(2/3))/(2*2^(2/3)*3^(5/6)*(-9*x^3 + Sqrt[-12*E^(
3*C[1])*x^3 + 81*x^6])^(1/3))}, {y[x] -> (-2*(-3*I + Sqrt[3])*E^C[1]*x + 2^(1/3)
*3^(1/6)*(-1 - I*Sqrt[3])*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x^6])^(2/3))/(2
*2^(2/3)*3^(5/6)*(-9*x^3 + Sqrt[-12*E^(3*C[1])*x^3 + 81*x^6])^(1/3))}}

Maple raw input

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

Maple raw output

[y(x) = 1/6/_C1*12^(1/3)*(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1/2))*
_C1^2)^(1/3)+1/6*x*12^(2/3)/(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1/2
))*_C1^2)^(1/3), y(x) = -1/12/_C1*12^(1/3)*(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x
^3-4)/_C1)^(1/2))*_C1^2)^(1/3)-1/12*x*12^(2/3)/(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1
^3*x^3-4)/_C1)^(1/2))*_C1^2)^(1/3)-1/2*I*3^(1/2)*(1/6/_C1*12^(1/3)*(x*(-9*x^2*_C
1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1/2))*_C1^2)^(1/3)-1/6*x*12^(2/3)/(x*(-9*x^2
*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1/2))*_C1^2)^(1/3)), y(x) = -1/12/_C1*12^
(1/3)*(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1/2))*_C1^2)^(1/3)-1/12*x
*12^(2/3)/(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1/2))*_C1^2)^(1/3)+1/
2*I*3^(1/2)*(1/6/_C1*12^(1/3)*(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)^(1
/2))*_C1^2)^(1/3)-1/6*x*12^(2/3)/(x*(-9*x^2*_C1+3^(1/2)*(x*(27*_C1^3*x^3-4)/_C1)
^(1/2))*_C1^2)^(1/3))]