n19 - Atsushi Inoue ECG 765: Mathematical Methods for...

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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 19 First-Order Difference Equations A. Particular Solutions and Complementary Functions Definition. Define the first-difference operator by y t = y t +1- y t Example. Consider y t = 2 . This can be written as y t +1 = y t + 2 . Because y 1 = y + 2 y 2 = y 1 + 2 = ( y + 2) + 2 = y + 2(2) y 3 = y 2 + 2 = ( y + 2(2)) + 2 = y + 3(2) the solution takes the form of y t = y + 2 t in general. Example. Consider y t =- . 1 y t which can be rewritten as y t +1 = 0 . 9 y t . 1 Because y 1 = 0 . 9 y y 2 = 0 . 9 y 1 = 0 . 9(0 . 9 y ) = (0 . 9) 2 y y 3 = 0 . 9 y 2 = 0 . 9(0 . 9) 2 = (0 . 9) 3 y the solution is y t = (0 . 9) t y in general. General Method. Consider y t +1 = ay t + c. The general solution is the sum of two components: a particular integral y p which is any solution of the difference equation y t +1 = ay t + c and a complementary function y c which is the general solution of y t +1 = ay t ....
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This note was uploaded on 10/22/2009 for the course ECG 765 taught by Professor Fackler during the Fall '07 term at N.C. State.

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n19 - Atsushi Inoue ECG 765: Mathematical Methods for...

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