I was considering this example:
Let $f: \mathbb{R} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2$ be given by $f(t, (x,y)) = (t \cos t + \sin t, t^2 \cos t + 2t \sin t)$ and let $\varphi(t)$ be the solution of $(x',y') = f(t,(x,y))$ with inicial condition $(x(0),y(0)) = (0,0)$. Then $\varphi(\pi) = \varphi(2\pi)$, for example, but $\varphi'(\pi)$ and $\varphi'(2\pi)$ are linearly independent. How can it be, if, by existence and uniqueness we would have that the solutions with inicial conditions $\varphi(\pi)$ and $\varphi(2\pi)$ are actually equal?
Your system is not autonomous, therefore ($\pi,(0,0))$ and $(2\pi,(0,0))$ are not equivalent initial data.