# Does this ODE have an equilibrium?

• explorer58
In summary, the conversation discusses the existence of an equilibrium in a system of connected reservoirs governed by a given ODE. The possibility of an extraneous solution is also mentioned. The conversation concludes with a request for insight on the conditions for equilibrium in the system.
explorer58

## Homework Statement

Two reservoirs are connected. Water drains from one reservoir to the other, governed by the following ODE:

dh/dt= -k1*(h)^0.5 -k2*(h-H)^0.5 , H<0, k1,k2>0

Does an equilibrium exist? What happens in terms of Picard's Existence Theorem? Draw a phase diagram of possible h* values.
Find an upper and lower bound for time it takes for reservoir to completely empty using respectively the maximum and minimum rates of decay of h on a well selected interval for h.

## The Attempt at a Solution

My friend and I are at odds. He says you can find a solution by saying

-k1*(h)^0.5 -k2*(h-H)^0.5=0
=> k1*h^0.5 = -k2*(h-H)^0.5

and then squaring both sides and solving for h. I on the other hand think that that will introduce an extraneous solution. It appears to me that, as it stands, the way it is set up, all terms are positive, and as such there is no equilibrium for this equation. However that's also just as confusing, because that would mean the rest of the question was a trick.

Any insight would be wholly appreciated.

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dh/dt stands for the rate of change of water in the system. If there is equilibrium, what condition must dh/dt satisfy?