Given plane autonomous system $\frac{dy}{dt} = Y(x,y), \frac{dx}{dt}=X(x,y)$
From this I can get a first order ODE $\frac{dy}{dx} = \frac{dy}{dt}\frac{dt}{dx}$, and we are given that $x(t)$ is $C^1$ and nonzero so we can say $\frac{dt}{dx} = \frac{1}{\frac{dx}{dt}}$. So we have $\frac{dy}{dx} = \frac{Y(x,y)}{X(x,y)}$.
Now the book I am reading says that these two ODEs are equivalent. I've shown one direction, but how do I get the other direction?