Does $\frac{\dot y}{\dot x}=\frac{dy}{dx}$ always work?

Question

Let consider the system: $$\begin{cases}\dot x=x\\ \dot y=-y+x^2.\end{cases}$$

To solve this system, my teacher made: $$\frac{dy}{dx}=\frac{dy}{dt}\frac{dt}{dx}=(-y+x^2)\frac{1}{dx/dt}=(-y+x^2)\frac{1}{x}.$$

And thus $y(x)=\frac{x^2}{3}+\frac{c}{x}.$

Question: Does this method always work? And if yes why?

Because I have the impression that it really simplify the problem, and I know that: $$\frac{dy}{dt}\frac{dt}{dx}=\frac{dy}{dx}$$ is not always true as this can show.
So why does it work here ?

Answer 1

It does always work, and it is called Chain Rule

Answer 2

If $y(x) = y(x(t))$, then by the chain rule

$$ \frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt} $$

Divide out $\frac{dx}{dt}$ from both sides.