The system of linear second-order differential equations m1x'l = -k1X1 + k 2(X2 - X1 )

Chapter 10, Problem 56

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The system of linear second-order differential equations m1x'l = -k1X1 + k 2(X2 - X1 ) describes the motion of two coupled spring/mass systems (see Figure 3.12.1). We have already solved a special case of this system in Sections 3.12 and 4.6. In this problem we describe yet another method for solving the system. (a) Show that (27) can be written as the matrix equation X "= AX ,where (b) If a solution is assumed of the form X= Ke"'1, show that X" = AX yields (A - ,\l)K = 0 where ,\ = <1i. (c) Show that if m1 = 1, m2 = 1, k1 = 3, and k2 = 2, a solution of the system is X = c (l)e it + c (l)e-it + c (-2)e Wit+ c (-2)e-Wit 1 2 22 3 1 4 1 . (d) Show that the solution in part (c) can be written as + b3(-D cos v6t + b4(-D sin v6t.