{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

This preview shows pages 1–3. Sign up to view the full content.

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 Δ y t = y t +1 - y t Example. Consider Δ y t = 2 . This can be written as y t +1 = y t + 2 . Because y 1 = y 0 + 2 y 2 = y 1 + 2 = ( y 0 + 2) + 2 = y 0 + 2(2) y 3 = y 2 + 2 = ( y 0 + 2(2)) + 2 = y 0 + 3(2) the solution takes the form of y t = y 0 + 2 t in general. Example. Consider Δ y t = - 0 . 1 y t which can be rewritten as y t +1 = 0 . 9 y t . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Because y 1 = 0 . 9 y 0 y 2 = 0 . 9 y 1 = 0 . 9(0 . 9 y 0 ) = (0 . 9) 2 y 0 y 3 = 0 . 9 y 2 = 0 . 9(0 . 9) 2 = (0 . 9) 3 y 0 the solution is y t = (0 . 9) t y 0 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 . Complementary Function: y t +1 = ay t . Let’s try a solution of the form y t = Ab t . Then the homogeneous difference equation becomes Ab t +1 = aAb t from which b = a . Thus,
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online