Economics Dynamics Problems 129

Economics Dynamics - Discrete dynamic systems 113 Hence c3 = y 0 y1 − c3 = c4 = r y1 − ry0 r Therefore the general solution satisfying the two

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Unformatted text preview: Discrete dynamic systems 113 Hence c3 = y 0 y1 − c3 = c4 = r y1 − ry0 r Therefore, the general solution satisfying the two initial conditions, is yn = y0 r n + y1 − ry0 r nr n Example 3.11 (equal real roots) Let yn+2 = 4yn+1 − 4yn This has the characteristic equation x2 − 4x + 4 = (x − 2)2 = 0 Hence, r = 2. yn = c3 (2)n + c4 n(2)n Suppose y0 = 6 and y1 = 4, then c3 = y0 = 6 4 − (2)(6) y1 − ry0 = = −4 c4 = r 2 Hence, the particular solution is yn = 6(2)n − 4n(2)n which tends to minus infinity as n increases, as shown in figure 3.14. In the case of the general solution yn = (c3 + c4 n)r n (1) (2) (3) If |r| ≥ 1, then yn diverges monotonically If r ≤ −1, then the solution oscillates If |r| < 1, then the solution converges to zero Figure 3.14. ...
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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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