P3 differential equations question

Please could someone help me with the following question:

An anthropologist is modelling the population of an Island. In the model the population at the start of the year t is P. The birth rate is 10 births per 1000 population per year. The death rate is m deaths per 1000 population per year.

a) show that dP/dt = (10-m)P/1000

b) At the start of year 0 the population was 108,000. Find an expression of P in terms of t

c) If the population is to double in 100 years, find the value m.

d) Explain why the population cannot double in less that 65 years.

Any help would be grately apprectiated.

Thanks!

Reply 1

Fermat

a)
Birth Rate = dB/dt = 10*(P/1000)
Death Rate = dD/dt = -m*(P/1000)

Population growth = dP/dt, and

dP/dt = dB/dt + dD/dt
dP/dt = (10-m)P/1000
=================

Reply 2

*hedgehog*

Thanks!

Reply 3

Fermat

b)
dP/dt = (10-m)P/1000 = kP,say wher k = (10-m)/1000

dP/P = k dt
lnP = kt + c
P = e^(kt + c)
P = e^kt.e^c
P = A.e^kt

at t = 0, P - 108,000, giving A = 108,000

P = 108,000.e^((10-m)t/1000)
=======================

Reply 4

Fermat

c)
at t = 100, P = 216,000 giving

216,000 = 108,000 e^100k
e^100k = 2
100k = ln2
100(10 - m)/1000 = ln2
10 - m = 10ln2
m = 10 - ln(1024)
m = 10 - 6.93
m = 3.07
=======

Reply 5

Fermat

d)
assume there are no deaths. Then death rate is zero => m = 0

let poulation double in T years.

2 = e^kT
kT = ln2
(10 - 0)/1000*T = ln2
T/100 = ln2
T = 100ln2
T = 100*0.693
T = 69.3 yrs
==========

It will take at least 69 yrs for the population to double, therefore population cannot double in lesss than 65 yrs

Reply 6

*hedgehog*

Cheers! You have been a great help!

Reply 7

greenie787

i've got another differential equation off a P3 paper, would be grateful if you could help and see if i'm going the right way about it.
Given that y = 0 when x = 0, solve the differential equation

dy/dx = (x + 1 + sinx)/cosy

here's what i've done:

∫ cosy dy = ∫ x + 1 + sinx dx

siny = x²/2 + x - cosx + C

using initial values C = 1 so,

siny = x²/2 + x - cosx + 1.
have i done it right up to now? if so what do i do next? thanks for any help