---
tag: 0WA7ZR95V4
kind: statement
role: proposition
name: Time translation of solutions of an autonomous system
msc: [34A34]
uses: [07BNRFQ0PJ, 04SW7WECS8]
sources:
  - url: https://en.wikipedia.org/wiki/Autonomous_system_(mathematics)
    locator: Properties
---

## 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|07BNRFQ0PJ]] $x' = f(x)$ 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.