- #1
Chris L T521
Gold Member
MHB
- 913
- 0
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]
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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]
