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Question 5 Suppose that A is the matrix (51 -1 A= 0 4 0 1 1...

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Question 5 Suppose that A is the matrix (51 -1 A= 0 4 0 1 1 3 (a) Find an invertible matrix P such that P-1AP = J where J is

Question 5 Suppose that A is the matrix (51 -1 A= 0 4 0 1 1 3 (a) Find an invertible matrix P such that P-1AP = J where J is a Jordan normal form. Now consider the system of differential equations f'(t) = 5f(t) +g(t) - h(t), 9'(t) = 49(t) and h'(t) = f(t)+g(t) + 3h(t). (b) Find the general solution to this system of differential equations. (c) What is the dominant long-term behaviour of this system? For what initial conditions will this long-term behaviour not occur? (d) Find the particular solution when f(0) = 1,90) = 0 and h(0) = 0.

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# 15 1 1 1 A-lo u o li i 3) 1 A-dil= 0 15- 01-11 1 1 1 3- . (5-d) (4-2) (3-2) + 1 (4-2) = 0 (4-2) I 22 -82 +15+1] = 0 (und) ((und) (2-4 = 0 d= 4,4,4 A for d = 4 (A-dI) x = 0 15-4 - 1 o 4-4 | My 1 1 3-4 lug Mit - Uz = 0 IO AM = GM= 3 of d= 4 Palol PAPo The vector corresponding to the 4t will be of long tim behaviour. - Take Cz = C₂ = 0 Now, f(t)=c, e g (t) = ci e Altc 462)

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