Question

Answer all questions (especially part b if unable to do part a)

QUESTION FOUR 25 MARKS] Consider the initial value problem d x()= A(0)x(t) +f(t), x(0) = xo where xo is some constant vector. A(T)dr A(t)= A(t)A()dr). Show that the matrix X (e) = ei A(0d (a) Assuming satisfies the matrix differential equation: Xt) = A(€)X(1) 10 Maris) (b) Obtain a solution to the initial value problem, given 0 A= f) x(0) -1 3 15 Marks

Answer 1

Gwon d t a) assuming S (A()dr) (pwr to shoco that mahx AC ALES A(t)d satisfies the D E. t KiH Ai X It) Frum Xi = ere AOdr Differen ha hig both side W.Y t tme we hare, S(A(dr exp phAmar LAlH A(T exp Air dt ATE CCuSing w) ACH) X(E) A ()dT proved exp (b T Find solu Hon fo IVp A = et xlo)() fiel= -3 fr dxAz TE +f1El . tt Ta king Laplace Transfom both side we have SXis)-6) = AY) t Fis) SLAIYIS)=Fis)+ t0) [iJ X IS1 s-3 [iT XIS S-3 -3/6-15-2) 2 (S-1)(5-2) S-)S-21 ' )s-2) Xis) S-5 st 35-2 (S22 (5-2) -23 S-3 (5-1) (S-2151) (sH) (S15-21 (SH) ($4S-2) S11(S-2) (5412 (5-2) + 4 (S-1)2 -4 Using Partal frachon eacpanston (S-1) S-2 2 (5-12

Waw, Tateiag Thwerse Laplace ensform 4+ et ai 2t 4C 5et 너{Vsg 2t 4etr 2tet-4e Fomula us ecl LE GE LixI= S here X(s) SKIS1-2ib) dt S-a -at teat

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