Consider a function $f:]a,b[\to\mathbb R$ satisfying the Lipschitz condition.
- Let $\varphi : ]\alpha , \beta[ \to ]a,b[$ be a maximal solution, $c\in\mathbb R$. Show the function $$\varphi ^ c: ]\alpha -c,\beta -c[ \to ]a,b[$$ $$ \varphi ^c (t):= \varphi (t+c)$$ is also a maximal solution.
- Let $\varphi,\psi$ be maximal solutions of $x'=f(x)$ with a common image. Prove one is a translation of the other.
For 1. I firstly proved "$\varphi$ is a solution $\implies$ $\varphi ^c$ is a solution". To prove it's also a maximal solution I tried the contrapositive:
Suppose $\varphi ^ c $ isn't a maximal solution. Then there exists an extension $\varphi _I^c$ of $\varphi ^c$ defined on an interval $I$ where $]\alpha - c,\beta -c[\subset I$. Now I need to build a function $\varphi_I$ defined on the interval $I+c$ such that $\varphi_I (t)$=$\varphi (t)$ for all $t\in ]\alpha,\beta[$ to conclude but it's not clear to me how should I do this.
For the second question, I don't understand the intuition behind it. If both solutions have a common image then there exists $t,s$ such that $\varphi(t)=\psi (s)$. Rewriting we have $\varphi(t)=\psi (t+c)$ where $c = s-t$. I don't know where to go from here. Any help is appreciated.