# Autonomous differential equation

I just need some help with this problem. Thank you.

"A pond forms as water collects in a conical depression of radius $$a$$ and depth $$h$$. Suppose that water flows in at a constant rate $$k$$ and is lost through evaporation at a rate proportional to the surface area.

(a) Show that the volume $$V(t)$$ of water in the pond at time $$t$$ satisfies the differential equation

$$\frac{dV}{dt} = k - \alpha \pi \left( \frac{3a}{\pi h} \right) ^{2/3} V ^{2/3}$$

where $$\alpha$$ is the coefficient of evaporation.

(b) Find the equilibrium depth of water in the pond. Is the equilibrium asymptotically stable?

(c) Find a condition that must be satisfied if the pond is not to overflow."

My work:

(a)

Consider the following

$$\frac{dV}{dt} = k - \alpha \pi \underbrace{\left( \frac{3a}{\pi h} \right) ^{2/3} V ^{2/3}} _{r^2}$$

where $$r$$ is the radius of the pond. Thus, we have

$$r^2 = \left( \frac{3a}{\pi h} \right) ^{2/3} V ^{2/3}$$

$$r^3 = \frac{3aV}{\pi h}$$

$$V = \frac{1}{3} \pi r^2 \left( \frac{hr}{a} \right) = \frac{1}{3} \pi r^2 L$$

which satisfies the differential equation.

(b)

$$\frac{dV}{dt} = 0$$

$$k - \alpha \pi r^2 = 0$$

$$r = \sqrt{\frac{k}{\alpha \pi}}$$

Then, the equilibrium depth of water in the pond is

$$L = \frac{h}{a} \sqrt{\frac{k}{\alpha \pi}}$$

In this particular case, I don't know how to show whether or not it is asymptotically stable.

(c)

$$L = \frac{h}{a} \sqrt{\frac{k}{\alpha \pi}} \leq h$$

Any help is highly appreciated.

An equilibrium depth of water in the pond implies that there is an equilibrium volume. Thus, a direction field can show whether or not it is asymptotically stable, which is what I have at

http://mygraph.cjb.net/

Here are the (random) values that I used to plot it:

$$\left\{ \begin{array}{ll} k = 1 \\ \alpha = 0.6 \\ a = 0.3 \\ h = 0.5 \end{array} \right.$$

Based on this information, I'd say the equilibrium is asymptotically stable.

Thanks anyway