Covariance of Newton's equation of motion under Galilean transformations

I’m reading Arnold’s mathematical methods of classical mechanics and in the section he talks about newton’s equation ($ddot{boldsymbol{x}}=boldsymbol{F}(boldsymbol{x}, boldsymbol{dot{x}},t)$) he says that this is invariant under Galilean transformations.
Right, I understand this, but now i wanna show mathematically that this happens

-For covariance under rotations assuming that i already now that the left hand side of newton’s equation does not depend explicitly on time: let $boldsymbol{x}:{phi}(t)$ be a solution for $ddot{boldsymbol{x}}=boldsymbol{F}(boldsymbol{x}, boldsymbol{dot{x}},t)$. Then if it’s invariant under rotations, a rotation $boldsymbol{chi}=Gboldsymbol{x}$ is also a solution. So it satifies
$$boldsymbol{ddot{chi}}=boldsymbol{F}(boldsymbol{chi},boldsymbol{dot{chi}})=Gboldsymbol{ddot{x}}=Gboldsymbol{F}(boldsymbol{x},boldsymbol{dot{x}})$$
Therefore,
$$boldsymbol{F}(boldsymbol{chi},boldsymbol{dot{chi}})=Gboldsymbol{F}(boldsymbol{x},boldsymbol{dot{x}})
implies boldsymbol{F}(Gboldsymbol{x},Gboldsymbol{dot{x}})=Gboldsymbol{F}(boldsymbol{x},boldsymbol{dot{x}})$$
and that’s the condition that Arnold showed. As i said, i assumed that it was already shown that the function $boldsymbol{F}$ has no explicit dependence on time.

HERE IS MY QUESTION: how do i show, in a similar way as i did to rotations , that in fact $boldsymbol{F}$ does not depend on time in a time translation $t rightarrow t+S$? if i do it, i think i’ll be able to do the same for space translation. Otherwise, if i’m not right in my approach please tell what i did wrong.