An equation like $y^{\prime \prime} + 2 y^{\prime} + y = 0$ has repeated roots: The characteristic polynomial is $r^2 + 2r + 1$ which has repeated roots $(-1,-1)$. Two basic solutions of the ODE are then $e^{-t}$ and $t e^{-t}$. More general examples exist with systems of ODE's and involve real Jordan form (aka real cannonical form) as explained in the textbook by Hirsch and Smale. My question is whether this has real applications.

In order to get repeated roots, the "true" ODE would have to be exactly a certain way and any slight deviation would destroy the repeated roots. I suppose people call non-repeated roots "generic". So this would seem very rare. Does this come up real applications? I could maybe see it in physics or somethings where there are well-defined laws governing a system but hard to see it in a setting where the ODE is really just a model of what is in reality a more complex system.



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