Centering y" = -y' in Example 3 will produce Yn+1 - 2Yn + Yn- 1 = -(6.t)2Yn. This can be
Chapter 6, Problem 28(choose chapter or problem)
Centering y" = -y' in Example 3 will produce Yn+1 - 2Yn + Yn- 1 = -(6.t)2Yn. This can be written as a one-step difference equation for U = (y, Z): Yn+1 = Yn + 6.t Zn Zn+l = Zn - 6.t Yn+l [ 1 0] [ Yn+1 ] = [1 6.t] [ Yn ] 6.t 1 Zn+l 0 1 Zn Invert the matrix on the left side to write this as U n+l = AU n. Show that det A = 1. Choose the large time step 6.t = 1 and find the eigenvalues Al and A2 = Al of A: A = [_! ~] has 1111 = 1121 = 1. ShowthatA6 is exactly I. After 6 steps to t = 6, U 6 equals U o. The exact y = cos t returns to 1 at t = 2rr.
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