# coordinate transformation in differential equation

I'm confused about coordinate transformations. What I understand is, that if we have $$f(x(t))=g(x(t)),$$ that we can write for each (bijective) $$h(x)$$ $$f\circ h(x(t))=g\circ h(x(t)).$$ If we can solve this, then we get an $$x(t)$$, such that $$h^{-1}\circ x(t)$$ is a solution in the old coordinates. In the case of differential equation, I'm confused though. Say we have $$\dot x(t)=x(t).$$ Now it seems that we can't speak of two functions that are equal (like $$f$$ and $$g$$ in our previous example), but it seems to me that we can still take a (diffeomorphism) $$h(x)$$, but I wouldn't know how that would work. You can't just composite each side with $$h(x)$$ like I did in the first example.

I hope my confusion is clear and that someone can help me out!