Can You Solve This Week's Problem? Hint: Use Gronwall's Inequality

[Chris L T521]

Here's this week's problem.

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Problem: Let $g:\Bbb{R}\rightarrow\Bbb{R}$ be Lipschitz and $f:\Bbb{R}\rightarrow\Bbb{R}$ be continuous. Show that the system
\[\left\{\begin{aligned} x^{\prime} &= g(x) \\ y^{\prime} &= f(x)y\end{aligned}\right.\]
has at most one solution on any interval for a given initial value.

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Hint: [sp]Use Gronwall's inequality. [/sp]

 

[Chris L T521]

No one answered this week's problem.

Now about this week's solution...it's rather embarrassing, but I'm still working on it ... (Headbang)

It looked rather easy to me when I picked it, but then for some reason I'm hitting a roadblock on figuring it out (lesson learned: never underestimate the difficulty of a Hirsch-Smale problem). I'll sleep on it tonight and hope to post a solution to this as soon as I possibly can (sometime later today). (Sweating)