How to Identify a homogeneous first order first degree ODE

Question

The following equation is homogeneous edit: y dx - x dy + 3x^2y^2e^(x^3) dx = 0 (source: Wolfram alpha) but it is not of the form of $f(zx,zy)= z(f(x,y))$. How do I identify such type of special Homogeneous equation that do not fit the definition?

Answer 1

Substituting $t=xy $ we have $y'= \left(\frac{t}{y}\right)' =\frac{t'x -t}{x^2 }$ thus we have $$\frac{t'x -t}{x^2 } =y' =\frac{dy}{dx} =\frac{y -y^3 x^2}{x+ x^3 y^2} =\frac{\frac{t}{x} -\frac{t^3}{x}}{xt^2 +x} =\frac{1}{x^2}\cdot \frac{t-t^3}{t^2 +1}$$ hence $$t'x -t =\frac{t-t^3}{t^2 +1}$$ and $$t'x =\frac{t-t^3 +t +t^3}{t^2 +1} =\frac{2t}{1+t^2} $$ therefore $$\left( \frac{1}{2t} +\frac{1}{2} t\right)dt =\frac{dx}{x}$$ so the solution is $$\frac{1}{2}\ln |t| +\frac{1}{4} t^2 = \ln |x| +C$$