4.14.12 \(x \left (6 y(x)^2+x\right ) y'(x)-3 y(x)^3+x y(x)=0\)

ODE
\[ x \left (6 y(x)^2+x\right ) y'(x)-3 y(x)^3+x y(x)=0 \] ODE Classification

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

Book solution method
Exact equation, integrating factor

Mathematica
cpu = 0.035346 (sec), leaf count = 64

\[\left \{\left \{y(x)\to -\frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\right \},\left \{y(x)\to \frac {\sqrt {x} \sqrt {W\left (\frac {6 e^{3 c_1}}{x^3}\right )}}{\sqrt {6}}\right \}\right \}\]

Maple
cpu = 0.013 (sec), leaf count = 30

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

DSolve[x*y[x] - 3*y[x]^3 + x*(x + 6*y[x]^2)*y'[x] == 0,y[x],x]

Mathematica raw output

{{y[x] -> -((Sqrt[x]*Sqrt[ProductLog[(6*E^(3*C[1]))/x^3]])/Sqrt[6])}, {y[x] -> (
Sqrt[x]*Sqrt[ProductLog[(6*E^(3*C[1]))/x^3]])/Sqrt[6]}}

Maple raw input

dsolve(x*(x+6*y(x)^2)*diff(y(x),x)+x*y(x)-3*y(x)^3 = 0, y(x),'implicit')

Maple raw output

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