# Find the equilibrium solution for an autonomous equation.

1. Aug 2, 2011

### clik80clak

1. The problem statement, all variables and given/known data

Consider a cylindrical water tank of constant cross section A. Water is pumped into the tank at a constant rate k and leaks out through a small hole of area a in the bottom of the tank. From Torcelli’s principle in hydrodynamics, it follows that the rate at which water flows through the hole is (alpha)(a)squareroot((2)(g)(h)) , where h is the current depth of water in the tank, g is the acceleration due to gravity, and alpha is a contraction coefficient that satisfies 0.5 < alpha < 1.0.

1. Show that the depth of water in the tank at any time satisfies the equation
dh/dt = ([k] - [(alpha)(a)squareroot{(2)(g)(h)}])/A
2. Determine the equilibrium depth he , of water, and show that it is asymptotically stable. Observe that he does not depend on A.

2. Relevant equations

We covered how to solve a DE using the integrating factor method after putting the eq. in to standard form but now we have moved on to autonomous eq.'s and I'm a little unsure how to go about solving this.

The problem looks more complex than it is because of all the brackets I had to use but I think it is pretty straight forward for someone familiar with the subject.

3. The attempt at a solution

1.

dh/dt = rate in - rate out

= (volume in/min)(1/area) - (volume out/min)(1/area)
= [k(m3/min) / A(m2)] - [((alpha)(a)squareroot{(2)(g)(h)}(m3/min) / A(m2)]
= ([k] - [(alpha)(a)squareroot{(2)(g)(h)}])/A

(Seems straight forward have I done this wrong?)

2.

dh/dt = m(h) h

=> [(k/hA - (alpha)(a)squareroot{(2)(g)}/sqrt{h}A] h

(Skipped the intermediate steps. Is this the right form? What do I do now?)