Doing the inverse of dy/dx = y(x) gives dx/dy = x(y)?

[lament bridge]

Hi! This is a question I was wondering about and I am not sure if true

If so, what would be the purpose? I have a teacher that solves certain differential equations doing this and i'm not sure why we can't simpl do it without the need to do the inverse..

I was also wondering if this could be a potential question where we ask to find the x(y)-inverse function of the solution y(x)?

[rapid ocean]

Sometimes it isn't possible to find y(x) in elementary functions, but it is possible to find x(y).
For example:
y' = y^2 (1 - y^2)
The solution of this equation (apart from y = 0 and y = ±1) is:
x = C + 1/y + (1/2)ln((1 - y)/(1 + y))
You can't express y from there using elementary functions.

[lament bridge]

Thank you as always!

[native magnet]

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