I am studying "Ordinary Differential Equations and Dynamical Systems" written by Gerald Teschl and I found a below proposition in this book.

(proposition) Consider a first order autonomous equation $x' = f(x)$ with $x(0) = x_0$, assume that $f$ is in $C(\Bbb R)$ (i.e., $f$ is continuously differentiable on $\Bbb R$)

If we have a solution $Φ(t)$ with $Φ(0)=x_0$, then the solution $Ψ(t)$ with $Ψ(t_0)=x_0$ is given by $Ψ(t)= Φ(t - t_0)$.

I can't prove this proposition. Can you give me an answer for this?

  • $\begingroup$ What exactly have you tried? You have one function given as solution of an IVP and are asked to confirm that a modified function is a solution of a modified IVP. This uses heavily that the ODE is autonomous. $\endgroup$ – LutzL Feb 26 '18 at 17:38
  • $\begingroup$ Try substituting $\Psi$ in the differential equation, using, perhaps, the chain rule. $\endgroup$ – rafa11111 Feb 26 '18 at 17:42

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