solving for the DE for $y$

Question

Let \begin{equation} y=\frac{dx}{dt}\frac{1}{x} \end{equation} How do i solve it? I tried to do it by separable variables but having 3 variables confuses me. Maybe when integration i can evaluate in the interval $[0,t]$?

Answer 1

If $y$ is a constant, $$yt+c=\log x.$$

If $y$ is a function of $t$,

$$\int y(t)\,dt=\log x.$$

If $y$ is a function of $x$,

$$t=\int\frac{dx}{x\,y(x)}.$$

If $y$ is a known function of both, you are stuck.

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But if you need to solve for $y$, all this is useless.