difference_eq - Notes on Difference Equations Definition...

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Unformatted text preview: Notes on Difference Equations Definition 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 ) Examples 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 ...
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This note was uploaded on 10/12/2011 for the course MATH 464 taught by Professor Dougbullock during the Fall '08 term at Boise State.

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