population growth of mosquitoes

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shakalandro

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So the problem that I have is that a population of mosquitoes grows in numbers at a rate proportional to the current population. They double every week, but die at a rate of 20000 a day. With a starting population of 200000, find an equation that models their growth. I was thinking that it was something like dP/dt = 2[sup:1dbortw9]t[/sup:1dbortw9]P - 20000(7t), and then you solve the differential equation and substitute in the initial value, but this ends up being nothing like the answer in the book which is P = 201977.31 - 1977.31e[sup:1dbortw9](ln2)t[/sup:1dbortw9]
 
Hello, shakalandro!

I haven't run through the problem yet, but there is a problem with their answer . . .

A population of mosquitoes grows in numbers at a rate proportional to the current population.
They double every week, but die at a rate of 20000 a day.
With a starting population of 200000, find an equation that models their growth.

I was thinking that it was something like:
. . . \(\displaystyle \frac{dP}{dt} \:=\:2^tP - 20000(7t)\)
and then you solve the differential equation and substitute in the initial value,
but this ends up being nothing like the answer in the book which is:
. . . \(\displaystyle P \:=\:201977.31 - 1977.31e^{(\ln2)t}\)
Click to expand...

I'm surprised that a Book left its answer like that . . .

\(\displaystyle \text{You see: }\:e^{(\ln 2)t} \;=\;\left(e^{\ln 2}\right)^t \;=\;2^t\)

 
Well I knew that, but that isn't exactly a problem, they just didn't solve it to its most elementary form. Do you think perhaps that the diff eq should have t represent days instead of weeks, would that make it come out right? Another problem with my answer is that when finding mu, the integral of 2[sup:2y225jgd]t[/sup:2y225jgd] is pretty nasty
 
As I see it the ODE should be:

\(\displaystyle \frac{dP}{dt} \, = \, \frac{ln(2)}{7}\cdot P \, - \, 20000\)
 
That doesn't seem to work, I got

P = 20000t + Ce[sup:18tviwj5]-ln7/2[/sup:18tviwj5]

regardless of what C is, this does not match the model that is necessary, their needs to be a 2[sup:18tviwj5]t[/sup:18tviwj5] somewhere in the end


EDIT! : wait, I miscalculated, this may be right

EDIT2!: Yes, I believe your ODE is the answer, P does not match the answer in the book, but considering the strange way they left the equation I am inclined to assume that they forgot the /7 in the exponent

P = 201977.31 - 1977.31e[sup:18tviwj5](ln2/7)t[/sup:18tviwj5]
 
shakalandro said:
That doesn't seem to work, I got

P = 20000t + Ce[sup:smgzkhs5]-ln7/2[/sup:smgzkhs5] <<< This is incorrect - how did you get that? Have you studied ODE and integrating factors?

regardless of what C is, this does not match the model that is necessary, their needs to be a 2[sup:smgzkhs5]t[/sup:smgzkhs5] somewhere in the end


EDIT! : wait, I miscalculated, this may be right

EDIT2!: Yes, I believe your ODE is the answer, P does not match the answer in the book, but considering the strange way they left the equation I am inclined to assume that they forgot the /7 in the exponent

P = 201977.31 - 1977.31e[sup:smgzkhs5](ln2/7)t[/sup:smgzkhs5]
Click to expand...
 
Yeah, I dunno, I forgot to multiply the integral in the integrating factor by t. I got it right in the end, read the edits.
 
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