Question

Chapter 7, Section 7.8, Question 010 Find the general solution of the given system of equations. X= 6 1) X -25 16 X-C 111 + C2 111 X=1 + 2 <= [(5)2 + (). X-Cil te 111 + C2 , 11t+cal X=1 + cz[(5)*e*** (9)e XC 11t+C2 Itellt 5

Answer 1

If the Coefficient matrix A has repeated eigen values. Then X1=Ve^($\lambda$t) is one solution and the second solution is of the form X2=(tV+W)e^($\lambda$t) where W is obtain by the system (A-$\lambda$I)W=V.
Thus the legenvalue 1 -11 with Algebraic multiplicity 2 has only one e zen veelor which is incomplete To find another independent eigenrector, we get a Second independent soluhin is of the form X₂ = (tv+w) Where W satisfies ellt (A-111) W-V Now, 7-355) w- (5 this equation many souhon, we choose, W-CO). Now, the and solution has of the form X₂ = elt/tvtw) ult 2 e 5 tt elle [(5) + (0)] Thus the general solution is, X(t) = GX,Lt) + X(t) = 9(5) 2" + 9 [16]te":(8) e")