4.14.9 \(x^3+3 x \left (y(x)^2+x\right ) y'(x)-3 x y(x)-2 y(x)^3=0\)

ODE
\[ x^3+3 x \left (y(x)^2+x\right ) y'(x)-3 x y(x)-2 y(x)^3=0 \] ODE Classification

[_rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.0242262 (sec), leaf count = 362

\[\left \{\left \{y(x)\to \frac {\sqrt [3]{c_1 x^2+\sqrt {x^3 \left (-2 c_1 x^2+c_1^2 x+x^3+4\right )}-x^3}}{\sqrt [3]{2}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{c_1 x^2+\sqrt {x^3 \left (-2 c_1 x^2+c_1^2 x+x^3+4\right )}-x^3}}\right \},\left \{y(x)\to \frac {i \sqrt [3]{2} \left (\sqrt {3}+i\right ) \left (c_1 x^2+\sqrt {x^3 \left (-2 c_1 x^2+c_1^2 x+x^3+4\right )}-x^3\right ){}^{2/3}+\left (2+2 i \sqrt {3}\right ) x}{2\ 2^{2/3} \sqrt [3]{c_1 x^2+\sqrt {x^3 \left (-2 c_1 x^2+c_1^2 x+x^3+4\right )}-x^3}}\right \},\left \{y(x)\to \frac {\left (2-2 i \sqrt {3}\right ) x-i \sqrt [3]{2} \left (\sqrt {3}-i\right ) \left (c_1 x^2+\sqrt {x^3 \left (-2 c_1 x^2+c_1^2 x+x^3+4\right )}-x^3\right ){}^{2/3}}{2\ 2^{2/3} \sqrt [3]{c_1 x^2+\sqrt {x^3 \left (-2 c_1 x^2+c_1^2 x+x^3+4\right )}-x^3}}\right \}\right \}\]

Maple
cpu = 0.024 (sec), leaf count = 21

\[ \left \{ x+3\,{\frac {y \relax (x ) }{x}}+{\frac { \left (y \relax (x ) \right ) ^{3}}{{x}^{2}}}+{\it \_C1}=0 \right \} \] Mathematica raw input

DSolve[x^3 - 3*x*y[x] - 2*y[x]^3 + 3*x*(x + y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((2^(1/3)*x)/(-x^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[
1]^2)])^(1/3)) + (-x^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])
^(1/3)/2^(1/3)}, {y[x] -> ((2 + (2*I)*Sqrt[3])*x + I*2^(1/3)*(I + Sqrt[3])*(-x^3
 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(2/3))/(2*2^(2/3)*(-x
^3 + x^2*C[1] + Sqrt[x^3*(4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(1/3))}, {y[x] -> (
(2 - (2*I)*Sqrt[3])*x - I*2^(1/3)*(-I + Sqrt[3])*(-x^3 + x^2*C[1] + Sqrt[x^3*(4 
+ x^3 - 2*x^2*C[1] + x*C[1]^2)])^(2/3))/(2*2^(2/3)*(-x^3 + x^2*C[1] + Sqrt[x^3*(
4 + x^3 - 2*x^2*C[1] + x*C[1]^2)])^(1/3))}}

Maple raw input

dsolve(3*x*(x+y(x)^2)*diff(y(x),x)+x^3-3*x*y(x)-2*y(x)^3 = 0, y(x),'implicit')

Maple raw output

x+3*y(x)/x+y(x)^3/x^2+_C1 = 0