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I am reading Hirsch/Smale/Devaney's "Differential equations, Dynamical Systems and an Introduction to Chaos", here:

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Why does this procedure shows there are no more solutions? I guess, specifically: Why they suppose $u(t)$ to be a solution and the compute the derivative of $u(t)e^{-at}$ instead of computing the derivative of $u(t)$? It seems there is some assumption that other solutions must be of that form but I don't see why this is true.

My guess is that due to $u(t)e^{-at} = \frac{u(t)}{e^{at}}$, if there was other solution different of the one we have, then it would be something like $u(t)=e^{at}v(t)$ for some $v(t)$ and this would cancel the $e^{at}$ but it's not entirely clear to me why all other solutions must be of that form.

Red Banana
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    Note that since the derivative of $u(t)e^{-at}$ is zero, you have $u(t)e^{-at}=C$ for some constant $C$, so $u(t)=Ce^{at}$. – csch2 Jan 16 '21 at 01:18
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    Related: https://math.stackexchange.com/questions/58097/prove-that-c-expx-is-the-only-set-of-functions-for-which-fx-fx – Hans Lundmark Jan 16 '21 at 10:42

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