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Differential equation for draining a pool

Vishak

New member
Oct 27, 2013
18
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.

Differential equation.jpg
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I have moved this topic to our Differential Equations sub-forum, as it is a better fit.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
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
Oct 27, 2013
18
Ok, now I've got this:

\(\displaystyle \frac{dV}{dD}= 100\)

\(\displaystyle \frac{dV}{dt} = \frac{dV}{dD}\cdot \frac{dD}{dt}\)

Leading to:

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

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
Looks good. :D
 

Vishak

New member
Oct 27, 2013
18
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

Administrator
Staff member
Feb 24, 2012
13,775
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
Oct 27, 2013
18
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

Administrator
Staff member
Feb 24, 2012
13,775
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
Oct 27, 2013
18
I'm guessing that for k > 2, it will overflow and for 0 <or= k < 2 it will empty?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I'm guessing that for k > 2, it will overflow and for 0 <or= k < 2 it will empty?
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 $k=0$.

In other words, in order for the pool to completely empty, there can be no water flowing into the pool, given that the initial amount of water is greater than zero. If there is any water flowing into the pool, no matter how slowly, then the equilibrium point will be greater than zero. Because the rate at which the water leaks out varies as the square root of the depth, as the level of the water decreases, the rate at which it leaks out will decreases as well, until at some point the rate at which it leaks approaches the rate at which it is being pumped in, and we approach equilibrium.
 

Vishak

New member
Oct 27, 2013
18
Thank you so much for that explanation! I understand it now :)