4.14.46 \(x \left (2 x^3+y(x)^3\right ) y'(x)=y(x) \left (2 x^3-x^2 y(x)+y(x)^3\right )\)

ODE
\[ x \left (2 x^3+y(x)^3\right ) y'(x)=y(x) \left (2 x^3-x^2 y(x)+y(x)^3\right ) \] ODE Classification

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

Book solution method
Homogeneous equation

Mathematica
cpu = 0.0291453 (sec), leaf count = 370

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

Maple
cpu = 0.017 (sec), leaf count = 27

\[ \left \{ 2\,{\frac {x}{y \relax (x ) }}-{\frac { \left (y \relax (x ) \right ) ^{2}}{2\,{x}^{2}}}-\ln \relax (x ) -{\it \_C1}=0 \right \} \] Mathematica raw input

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

Mathematica raw output

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

Maple raw input

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

Maple raw output

2*x/y(x)-1/2/x^2*y(x)^2-ln(x)-_C1 = 0