Time translation of solutions of an autonomous system

Statement

Let $U$ be an open subset of $\mathbb{R}^n$, let $f : U \to \mathbb{R}^n$ be a function. Let $\gamma : I \to U$ be a solution of the autonomous system $x' = f(x)$ on an interval $I$. Let $c \in \mathbb{R}$ and let $I - c = \{t - c : t \in I\}$. Then the function $\delta : I - c \to U$ given by $\delta(t) = \gamma(t + c)$ is also a solution of $x' = f(x)$.

Remarks

Proof: by the chain rule, $\delta'(t) = \gamma'(t + c) = f(\gamma(t + c)) = f(\delta(t))$. So it is enough to solve an initial value problem with initial time $0$. The solution with $x(t_0) = x_0$ is $t \mapsto \varphi(t - t_0)$, where $\varphi$ solves $x(0) = x_0$. The result fails for non-autonomous equations. For $x' = t$, the solution $t^2/2$ is a solution, but $(t+1)^2/2$ is not.

Explanations

L1 human

If the law of motion does not change with time, a motion that starts later is the same motion, only delayed. So shifting a solution in time gives another solution.