S.O.S. Mathematics CyberBoard

Hey, if someone could please help me out with this problem I'd really appreciate it!


1. The birth rate in a particular state is 2% per year and the death rate is 1.3% per year. The current population of the state is 8,000,000.

a) Write a differential equation which models the population of the state. Be sure to define all variables.
b) Solve your differential equation.
c) How long will it take the population to reach 10,000,000.
d) Generate a graph which shows the long term (50 years) population project of the state under these conditions.

2. Now in addition to the facts above, assume that people are moving out of the state at a constant rate of 60,000 people per year.

a) Write a differential equation which models the population of the state given this emigration.
b) Solve your differential equation.
c) Will the population ever reach 10,000,000? If so, when? If not, why not?
d) Generate a graph which shows the long term (50 years) population project of the state under these conditions.
e) At was constant rate will people have to leave the state in order for the state to have a constant population?

What have you tried? Where are you stuck?

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So far for #1 I've got (a) and (b) which is:

rate of birth = 2% =.02
rate of death = 1.3% =.013
rate of change of population = dx/dt = .007x
x = total population

∴lnx = .007t+c

t in years and c is a constant

solving further we get

x= 8000000*e.007t

(a) and (b) for #2 is:

dx/dt = .007x-60000
dx/.007x-60000 = dt
integrating we get
ln(.007x-60000)/.007 = t + c
c is a constant
.007x - 60000 = Ke0.007t

x= K1e0.007t+60000/.007

x= K1e0.007t+8571428.571

x(0)=8000000

==> K1=-571428.571

∴x = -(571428.571)e0.007

At least that's what I got so far.