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Problem of the Week #76 - November 11th, 2013

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Chris L T521

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Jan 26, 2012
995
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:


Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Chris L T521

Well-known member
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Jan 26, 2012
995
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)
 
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