# Differential Equations - Exponential Decay

## Homework Statement

We have the ODE y' = -ky + R for a population y(t) where death rate exceeds birth rate, counteracted by a constant restocking rate.
I'm assuming k is the decay constant and R is the restocking rate

The population at time t0 = 0 is y0, and I have to find a formula for y(t)
Also, interpret the solution in terms of the long term behavior (0< y0 < R/k, y0 = R/k, and y0 > R/k).

y' = -ky + R

## The Attempt at a Solution

I solved the DE and got y = R/k + Ce^(-kt)
But how do I know what this does in those given intervals of y0?
Also if k = .5, R = 2, how would I graph the solutions?

## Answers and Replies

Mark44
Mentor
You should get three different graphs for your solution, when you substitute y(0) = y0 into the solution you found. One solution is for y0 between 0 and R/k, one is for y0 = R/k, and one is for y0 > R/k.

Think about these in terms of replenishment rate < death rate, replenishment rate = death rate, and replenishment rate > death rate. How is the population affected in each case?

So...first interval - exponentially decreasing graph
when y = R/k, it's a constant, straight line along R/k?
and when y is big, it's increasing?

Mark44
Mentor
Sounds reasonable. Are these what you think, or does your solution back them up?

These are what I think
I'm not sure how to find graph solutions from my equation: y = 4 + Ce^(-.5t)

Thanks for your help!

Mark44
Mentor
Where do the k=.5 and R=2 come from? Are they part of the problem statement? You didn't mention these values there. I'm going to ignore them for the time being.

You are given that y(0) = y0, so substitute this into your solution equation to find C.
y(t) = R/k + Ce-kt
y(0) = y0

==> y0 = R/k + C
==> C = y0 - R/k

So the solution is
y(t) = R/k + (y0 - R/k)e-kt

Now look at this solution for the three different scenarios. Each one will have an effect on the sign of the exponential term. You're not going to get precise graphs, since you don't know the values of R and k, but you should get a good idea of the general behavior.