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QUESTION FOUR [25 MARKS] (a) Prove that if (t, to) is the transition matrix for the systemx(t) = A(t)x(t), then the unique so
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Solution A=Ca)h : EIR isRaid tobe an eigenvalve f Co Lumn Xin uh thot AXCH=XH) o Ca lumn xis Called tigen eect by ligen valueAvmayoneoue tyton Congider the nm 2 hene FI Where xt ret n trs) Conidey the Coroespornting hrmege neue sfomy AY Feldl=pcc herCeneider igen veebs Gnatpen d/se ElMd No -gen value atJAF amd 2 A-rA C ama e - Henie the geroa elution ut 2t 2t J k Xe Ct)=CYel4) owd have we out 2 2/f hene Fu e 2t 2 4t -eut 2 e 2 e 2t -e2t e2t ut e-44 2-024 2 -- jJr 2- Jdt 2 L 2t-e2t tP(f0-(F)P e2The generalolutm eatpe ef + Rgetf 4te4t (4c2 t4 (Gtd)(+C2-)1 eut -1)-() 1-(0+7) & C2 O (24ut) et +(o+ 24- 1)eut eXtut) e4t He

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