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Is it possible to find an analytical solution of the following ODE? $$y^\prime(x)=z(x)-a\cdot \min(y(x),b),~ x\geq 0$$ subject to a given condition $y(0)$, and $z(x)$ is a given function (smooth enough), and $a$ and $b$ are two given positive constants.

It is well known that we can find the solution of $y^\prime(x)=z(x)-a\cdot y(x),~ x\geq 0$. I am wondering how to solve the ODE with a "min" in it?

Thank you very much!

Mike HWANG
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  • The min is either $b$ or $y(x)$. In both cases there is an analytical solution, so it's possible. You have to deal with initial conditions where $y(x)$ goes through $b$. Depending on the behavior of $z$, it can get quite difficult though: you have to solve for $y(x)=b$, with $y$ the analytical solution on the preceding interval. – Jean-Claude Arbaut Sep 29 '21 at 07:03
  • I know it is not the same problem, but maybe this similar one can help https://math.stackexchange.com/a/3214722/399263. As JC. Arbaut said, initial conditions may play a big role in eliminating one of the two cases of the minimum, but this imply that you have informations about $z(0)$ at least. – zwim Sep 29 '21 at 09:05
  • I'm afraid the best you can do is case work. Do the case when $b\geq y(x)$ and when $b\leq y(x)$ and find the solution in both cases. Whether or not you will be able to unite both of these cases to yield a continuous solution is anyone's guess. – K.defaoite Sep 29 '21 at 10:11

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