I How to determine if x(t) is a solution to a system x'(t)=f(x)

[NicolaiTheDane]

For example having looked a solution sheet, I know that $x(t)=\sin(t)$ is not a solution for any system of the form $\dot{x}(t)=f(x)$. I assume this is rather simple, but I simply cannot get my head around why it wouldn't be. I'm guessing it has to do with the dependence on $x$ rather then $t$ but I haven't gotten anywhere.

 

[PeroK]

If $x(t) = \sin(t)$, then $x'(t) = \cos(t) = \pm \sqrt{1- \sin^2(t)} = \pm \sqrt{1 - x(t)^2}$