Question about differential equations and flows Consider the system of differential equations
$$ \begin{aligned} x' &= x\cos(tx) \\ y' &= y\sin(tx) \end{aligned} $$
How can I prove that the flow $\phi(·, t_0,(x_0, y_0))$ is $C^k$ for $k>1$, where $x(t_0) = x_0$, $y(t_0) = y_0$?
Also, it is true that $$\{\phi(t, 0,(1, 1)) : t \in\mathbb{R}\} \cap \{(x, y) \in\mathbb{R}^2
: x < 0\} = ∅?$$
Really, I don't have any idea about how to do it, sorry. Any ideas?

You only have to consider a new variable $t$. In the space $\mathbb R^3$ with coordinates $(t,x,y)$ you have the vector field
$$
(1, x\cos(tx), y\sin(tx))
$$
which corresponds to your non autonomous system. Now you only have to apply the fundamental theorem of flows to assure the existente of the requiered flow.
Is not a complete answer but maybe you can fill the details.
