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QUESTION FOUR Consider the initial value problem x(t) A(t)x(t) f(t), dt where xo is some constant vector (a) Show that the associated homogenous system, x(t) A(t)x(t), has its transi tion matrix as X(t)e Ar)dr provided AeJtr)-Ar) A(t) for all t 10 Marks A(t) A(T)dr (b) Obtain a solution to the initial value problem, given that ()- () 4 A = 6 et and x(0) 1 15 Marks f(t)

Answer 1

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matrix from sther dimenmian, hcy alt byAlt), ce which in 7 sct ALT)dt hen A CT) d7 dx lt) dt dt 上d KI dt t KO Who to 2(2) Naw (w) = 11 dt gemerah Att)dt A Lt)+ Al IM3 P字-2

A LL) All A (T) d7 hun implifier bo K dt KI t ALT)dt to A lt) e 1 = A lt)XLt). Hence tu porouf of parrt a) (4) fir part b) A = flt)= ) (et amd (0)=

e boramsitin matrix iven Ay t A CT) dT to Nerw 4t-o t -t 4t t 6t -t page-y 4t t