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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
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.
y∗ + ay∗ + by∗ = c
so long as 1 + a + b = 0.
yn+2 − 4yn+1 + 16yn = 26
y∗ − 4y∗ + 16y∗ = 26
y∗ = 2
Hence, yp = 2. The general solution is, then
yn = c1 4n cos πn
+ c2 4n
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.
- Fall '11