Economics Dynamics Problems 132

Economics Dynamics Problems 132 - 116 Economic Dynamics...

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Unformatted text preview: 116 Economic Dynamics 3.8.2 Nonhomogeneous A constant coefficient nonhomogeneous second-order difference equation takes the general form yn+2 + ayn+1 + byn = g(n) (3.20) If g(n) = c, a constant, then yn+2 + ayn+1 + byn = c which is the form we shall consider here. As with second-order differential equations considered in chapter 2, we can break the solution down into a complementary component, yc , and a particular component, yp , i.e., the general solution yn , can be expressed yn = yc + yp The complementary component is the solution to the homogeneous part of the recursive equation, i.e., yc is the solution to yn+2 + ayn+1 + byn = 0 which we have already outlined in the previous section. Since yn = y∗ is a fixed point for all n, then this will satisfy the particular solution. Thus y∗ + ay∗ + by∗ = c c y∗ = 1+a+b so long as 1 + a + b = 0. Example 3.13 yn+2 − 4yn+1 + 16yn = 26 Then y∗ − 4y∗ + 16y∗ = 26 y∗ = 2 Hence, yp = 2. The general solution is, then yn = c1 4n cos πn πn + c2 4n +2 3 3 Example 3.14 yn+2 − 5yn+1 + 4yn = 4 ...
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This note was uploaded on 12/14/2011 for the course ECO 3023 taught by Professor Dr.gwartney during the Fall '11 term at FSU.

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