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 = 0.189442 (sec), leaf count = 433

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

Maple
cpu = 0.026 (sec), leaf count = 47

\[ \left \{ \ln \left ({\frac {{x}^{2}-xy \relax (x ) + \left (y \relax (x ) \right ) ^{2}}{{x}^{2}}} \right ) +\ln \left ({\frac {x+y \relax (x ) }{x}} \right ) -2\,\ln \left ({\frac {y \relax (x ) }{x}} \right ) -\ln \relax (x ) -{\it \_C1}=0 \right \} \] 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),'implicit')

Maple raw output

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