i need to show that doesn't exist f: R->R C1 class, such that:
$\psi(t) = \cos(t), t \in \mathbb{R}$ is solution of:
$\frac{dx}{dt} = f(x) \;\;\;\; (1) $
My attempt:
Suppose that exists $f:R\rightarrow R$, C1 class ; $\psi(t) = \cos(t)$ is solution of (1) that is:
$\frac{d\psi}{dt} = f(\psi(t)) $.
but note that: $\frac{d^2\psi}{dt} = \psi(t) = f'(\psi(t))*\psi\;'(t)$
so we got:
$\psi\;'(t) = \frac{\psi(t)}{f'(\psi\;(t))} \; ; f'(\psi\;(t)) \neq 0\; \forall t \in \mathbb{R}$
then $f$ is monotonically increasing or decreasing.
so here i'd like to find some contradiction but i can't. Perharps i took the wrong way.
Thanks