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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!

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