This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Notes on Diﬀerence Equations
A diﬀerence equation is an equation in which the unknown quantity is a function, N (t), satisfying the rule
N (t + 1) − N (t) = some formula involving N
In more mathematical notation:
∆N = f (N )
1. ∆N = 0.03N
2. ∆N = 250N − N 2
5000 Properties of Solutions
1. Any diﬀerence equation that we study will have inﬁnitely many solutions. Example 1, above,
has solutions: N (t) = 50(1.03)t , N (t) = 100(1.03)t, N (t) = 150(1.03)t, and so on.
2. If you know one point of initial data, say N (0) = 100, then you can evolve a solution in
forward time. Working backwards can be a lot harder.
3. While it is not true that solutions can’t intersect, you can often learn a lot by pretending they
can’t intersect. This is especially true for equilibrium solutions.
4. A constant function that solves a diﬀerence equation is called an equilibrium solution. You
ﬁnd them by setting the right hand side equal to zero and solving for N .
5. In between equilibrium solutions, you can often predict the general behavior of all other
solutions. The trick is to think about whether solution graphs go up or down. And assume
that solutions cant cross. 1 ...
View Full Document