3. Show that costsin t (t) = sint cos t is a fundamental matrix solution of...
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3. Show that costsin t Þ(t) = sint cos t is a fundamental matrix solution of the nonautonomous linear system x = A(t)x with -2 cos² t -1-sin 2t] -2 sin² t in2²]. Find the inverse of (t) and use Theorem 1 and Remark 1 to solve the nonhomogenous linear system -[-2006²2 A(t) = x = A(t)x+ b(t) with A(t) given above and b(t) = (1, e-2)T. Note that, in general, if A(t) is a periodic matrix of period T, then corresponding to any fundamental matrix (t), there exists a periodic matrix P(t) of period 2T and a constant matrix B such that (t) = P(t)e Bt. Cf. [C/L], p. 81. Show that P(t) is a rotation matrix and B = diag[-2,0] in this problem. 3. Show that costsin t Þ(t) = sint cos t is a fundamental matrix solution of the nonautonomous linear system x = A(t)x with -2 cos² t -1-sin 2t] -2 sin² t in2²]. Find the inverse of (t) and use Theorem 1 and Remark 1 to solve the nonhomogenous linear system -[-2006²2 A(t) = x = A(t)x+ b(t) with A(t) given above and b(t) = (1, e-2)T. Note that, in general, if A(t) is a periodic matrix of period T, then corresponding to any fundamental matrix (t), there exists a periodic matrix P(t) of period 2T and a constant matrix B such that (t) = P(t)e Bt. Cf. [C/L], p. 81. Show that P(t) is a rotation matrix and B = diag[-2,0] in this problem.
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