The following homogeneous system of linear first-order differential equations x\{t) =

Chapter 9, Problem 4

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The following homogeneous system of linear first-order differential equations x\{t) = 5xi(f) - X2(t) +2x^1)+ x4(t) x'2(t) = -X|(r) +4*2(0 +2*4(0 x'3(t) = 2*1 (?) +4*3(/) + *4(0 ^(0= +(0 +2*2(0+ *3(0 +5*4(r) can be written in the matrix-vector form x'(0 = Ax(0, where -uC) 5-121 *2(0 and A -1 4 0 2 *3(0 2 0 4 1 *4(0 1 2 15 Constructing the general solution to the system of differential equations x(0 = Ciex,'\] + C2^A2'v2 + Ciek}'\3 + C4e''4 '\4 requires the eigenvalues A.], X2, X3, and X4 of A. Finding the eigenvaues of A using the QR method requires a symmetric tridiagonal matrix similarto A. Use Householder's method to find such a matrix.