Ordinary differential equations Proposition

Time translation of solutions of an autonomous system

Builds on Autonomous system Solution of an ordinary differential equation

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.

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Differential equation concept
Function concept
Interval concept
Ordinary differential equation definition
Solution of an ordinary differential equation definition
Autonomous system definition
Time translation of solutions of an autonomous system statement

Tag	`0WA7ZR95V4`
URI	`m2://0WA7ZR95V4`
Kind	statement, proposition
Writing	W0 stub
MSC	34A34
Status	This copy of the site has no build data. Run `m2 status 0WA7ZR95V4`.
Uses	Autonomous system, Solution of an ordinary differential equation

Time translation of solutions of an autonomous system

Statement

Remarks

Reading path

Further reading

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