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

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

[[_homogeneous, `class G`], _rational]

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.127114 (sec), leaf count = 359

\[\left \{\left \{y(x)\to \frac {e^{\frac {8 c_1}{3}}}{\sqrt [3]{3 \sqrt {3} \sqrt {-x^6 \left (e^{8 c_1}-27 x^8\right )}-27 x^7}}+\frac {\sqrt [3]{\sqrt {3} \sqrt {-x^6 \left (e^{8 c_1}-27 x^8\right )}-9 x^7}}{3^{2/3} x^2}\right \},\left \{y(x)\to \frac {i \sqrt [6]{3} \left (\sqrt {3}+i\right ) \left (\sqrt {3} \sqrt {-x^6 \left (e^{8 c_1}-27 x^8\right )}-9 x^7\right ){}^{2/3}-\left (\sqrt {3}+3 i\right ) e^{\frac {8 c_1}{3}} x^2}{2\ 3^{5/6} x^2 \sqrt [3]{\sqrt {3} \sqrt {-x^6 \left (e^{8 c_1}-27 x^8\right )}-9 x^7}}\right \},\left \{y(x)\to \frac {\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {3} \sqrt {-x^6 \left (e^{8 c_1}-27 x^8\right )}-9 x^7\right ){}^{2/3}}{x^2}+i \sqrt [3]{3} \left (\sqrt {3}+i\right ) e^{\frac {8 c_1}{3}}}{2\ 3^{2/3} \sqrt [3]{\sqrt {3} \sqrt {-x^6 \left (e^{8 c_1}-27 x^8\right )}-9 x^7}}\right \}\right \}\]

Maple
cpu = 0.021 (sec), leaf count = 33

\[ \left \{ \ln \relax (x ) -{\it \_C1}+{\frac {3}{8}\ln \left ({\frac { \left (y \relax (x ) \right ) ^{3}+2\,x}{x}} \right ) }-{\frac {3}{8}\ln \left ({y \relax (x ) {\frac {1}{\sqrt [3]{x}}}} \right ) }=0 \right \} \] Mathematica raw input

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

Mathematica raw output

{{y[x] -> (-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(1/3)/(3^(2/3)*x
^2) + E^((8*C[1])/3)/(-27*x^7 + 3*Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(1
/3)}, {y[x] -> (-((3*I + Sqrt[3])*E^((8*C[1])/3)*x^2) + I*3^(1/6)*(I + Sqrt[3])*
(-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(2/3))/(2*3^(5/6)*x^2*(-9*
x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*C[1]) - 27*x^8))])^(1/3))}, {y[x] -> (I*3^(1/3)*(
I + Sqrt[3])*E^((8*C[1])/3) + ((-1 - I*Sqrt[3])*(-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^
(8*C[1]) - 27*x^8))])^(2/3))/x^2)/(2*3^(2/3)*(-9*x^7 + Sqrt[3]*Sqrt[-(x^6*(E^(8*
C[1]) - 27*x^8))])^(1/3))}}

Maple raw input

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

Maple raw output

ln(x)-_C1+3/8*ln((y(x)^3+2*x)/x)-3/8*ln(1/x^(1/3)*y(x)) = 0