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#1 2011-09-30 22:34:44

viny
Guest

Differential equation

Any one give a hand on this please???


The population of butterflies ina butterfly house is being observed by its keeper. He knows that his butterflies will grow at a rate that is proportional to their current population adn that in the absence of any outside factors the population will double every 10 days. The keeper estimates that, on average, on any given day 8% of butterflies escape the house. The current population is 100 butterflies.

Will this population of butteflies survive? If yes, describe its growth pattern. If not, when will they die out?

You must use differential equations to model this problem.

#2 2011-09-30 23:04:12

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: Differential equation

hi viny

The population doubles every 10 days in the absence of outside factors. (I'm assuming that sneaking out the house (8%) is such a factor.)

So ignore it for the moment and just work on the doubling factor.

Call the population N and let 'k' be a constant of proportion.

Now you can do 'separation of variables' to get N in terms of t, and use the doubling factor to calculate k.

Have a go at that and post back with an answer or for more help. 

Now use that equation to find out how much the population goes up in one day (t = 1)

So if you begin with 100, how many will you have the next day?

And if 8% escape?

That should answer the question.

Bob

Last edited by Bob (2011-09-30 23:20:37)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2011-09-30 23:07:13

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Differential equation

Hi;


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#4 2011-10-01 12:49:46

viny
Guest

Re: Differential equation

This is the way I think
dP/dt proportional to kP.
P doubles everyday 10 days: 2 to the power of t/10 * P
8% escape each day: 0.08Pt

so my equation is dP/dt = 2 to the power of t/10 * P - 0.08Pt

What do you think?. I don't understand how you take the log of P and there is relationship to time

#5 2011-10-01 18:14:44

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Differential equation

Hi viny;

viny wrote:

so my equation is dP/dt = 2 to the power of t/10 * P - 0.08Pt

Is this what you think it is?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#6 2011-10-01 20:15:58

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: Differential equation

hi viny

The log arises from the differential equation (DE).

To integrate this,  the usual method is to 'separate the variables'.

Get the 'P' terms on one side and the 't' terms on the other.

(Some people worry here about splitting the differentials like this, but it works as long as you quickly put in some integral signs)


replacing e^C with another constant, 'A', we get

Once you substitute to find 'k' and put in the escape factor, you get bobbym's first equation

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#7 2011-10-01 21:12:20

Bob
Administrator
Registered: 2010-06-20
Posts: 10,053

Re: Differential equation

Some more thoughts on this:

Put t = 0     =>     P = A

Put t = 10    =>    P = 2A

so

See graph below.

With A = 100, you can see that after one day A is just over 107.

So if you take off 8% the population has gone down.

In fact you don't have to have A = 100.

If on the first day the population is 'A', then after a day the population is  = 1.071773A - 1.08A

So, on any day, the population is decreasing.

Bob

Last edited by Bob (2011-10-01 21:12:48)


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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