When can a non-autonomous system NOT be re-written as an autonomous system?

Given a non-autonomous system $x'=f(x,t)$, you can introduce new vector function $u(t)=(x(t),t)$ which satisfies the autonomous system $u'=g(u)$ with $g(u)=(f(u),1)$. So the answer is yes, you can always turn a system into autonomous.

The implication is that the dimension of the system goes up. And while autonomous systems are often easier to understand by analysis of their equilibria, we do not get a free lunch here: the new system $u'=g(u)$ has no equilibria ($g$ never vanishes).

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