Differential equation for draining a pool

Vishak

New member
Hi MHB. Can someone help me with this one please? I don't understand what the question is really saying...in particular part (c).

I tried to set up dD/dt = k - D^1/2 but it doesn't seem correct. Thanks.

MarkFL

Staff member
I have moved this topic to our Differential Equations sub-forum, as it is a better fit.

MarkFL

Staff member
Your answer to part (a) is not quite correct. We know:

(1) $$\displaystyle \frac{dV}{dt}=\text{flow in}-\text{flow out}$$

We are given:

$$\displaystyle V=100D,\,\text{flow in}=k,\,\text{flow out}=\sqrt{D}$$

So, substituting these given values into (1), what do we obtain?

Vishak

New member
Ok, now I've got this:

$$\displaystyle \frac{dV}{dD}= 100$$

$$\displaystyle \frac{dV}{dt} = \frac{dV}{dD}\cdot \frac{dD}{dt}$$

$$\displaystyle \frac{dD}{dt}= \frac{k-\sqrt{D}}{100}$$

Staff member
Looks good.

Vishak

New member
Okay, thanks

So now for part (b) I've got the only equilibrium solution:

$$\displaystyle D = k^{2}$$

Which is stable.

But I'm completely lost looking at part (c)

MarkFL

Staff member
Your equilibrium point is correct, and is stable since:

i) For $$\displaystyle D<k^2$$ we have:

$$\displaystyle \frac{dD}{dt}>0$$

ii) For $$\displaystyle D>k^2$$ we have:

$$\displaystyle \frac{dD}{dt}<0$$

which shows that for all values of $D$, we must have:

$$\displaystyle \lim_{t\to\infty}D(t)=k^2$$

So, for part (c) you want to look at what values of $k$ cause the equilibrium point to satisfy the condition that the pool overflows or empties. Where would these equilibrium points be?

Vishak

New member
Ok, thanks.

The thing that's still tripping me up is this part:

$$\displaystyle D_{0} \in (0,4)$$

And the fact that right now D0 isn't in the equation...

MarkFL

Staff member
Ok, thanks.

The thing that's still tripping me up is this part:

$$\displaystyle D_{0} \in (0,4)$$

And the fact that right now D0 isn't in the equation...
$$\displaystyle D_0=D(0)$$

and we are given that:

$$\displaystyle 0<D_0<4$$

Can you now state what values of $k$ will cause the pool to empty and overflow?

Vishak

New member
I'm guessing that for k > 2, it will overflow and for 0 <or= k < 2 it will empty?

MarkFL

You are correct that for $2<k$ the pool will overflow, since the equilibrium point will be greater than the depth of the pool.
In order for the pool to empty, we require the equilibrium point to be $D=0$, so we require $k=0$.