The following homogeneous system oflinear first-order differential equations *i(0

Chapter 9, Problem 10

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The following homogeneous system oflinear first-order differential equations *i(0 =-4.V|(r) - X2(0 x'2(t) = -*,(/) -4*2(0+ 2*3 *3(0 = 2*2(0 - 4*3(0 - ^4(0 *4(0 = -*3(0 + 4*4(0 can be written in the matrix-vector form x'(t) A\{t), where x(0 = to construct the general solution to the system of differential equations, x(f) = c\eXi'\\ + c1e'-1 '\2 + cye^'yy + cAe kA'vA, where c\, C2, C3, and C4 are arbitrary constants and Xi, X2, X3, and I4 are eigenvalues with the corresponding eigenvectors X], X2, X3, and X4. a. Use the QR method to find A-i,..., X4. b. Use the Inverse Power method to find X],..., X4. c. Form the general solution of x'(r) = Ax{t). d. Find the unique solution satisfying x(0) = (2, 1, 1, 3)