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n21 - Atsushi Inoue ECG 765 Mathematical Methods for...

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Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 21 Higher Order Linear Difference and Differential Equations We will consider second-order difference and differential equations. The dy- namics of solutions depend on the solution to some quadratic equations called characteristic equations. Outline A. Second-Order Difference Equations B. Higher-Order Differential Equations A. Second-Order Difference Equations Consider x t = ax t - 1 + bx t - 2 + y (1) A general solution for this difference equation is the sum of a particular solution to (1) and a complementary function that is a solution to the ho- mogeneous version of (1): x t = ax t - 1 + bx t - 2 . (2) A particular solution is any solution to (1). If a + b 6 = 1, for example, try x t = x t - 1 = x t - 2 = x : x = ax + bx + y or x = 1 1 - a - b y Suppose that the complementary function takes the form of x t = Ak t . Then x t - ax t - 1 - bx t - 2 = Ak t - aAk t - 1 - bAk t - 2 = 0 1
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If A 6 = 0 and k 6 = 0, then k 2 - ak - b = 0 .
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