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Name________________________ Student I.D.___________________ Math 2250-1 Quiz 12 Solutions November 30, 2012 Here is a matrix, and its eigenvalue-eigenvector data: K3 A := For l1 =K1, v = 2 2 K3 1 1 . For l2 =K5, v = 1 K1 . 1) Consider the following first order system of first order differential equations, which could arise from an input-output model: x1 # t =K3 x1 C 2 x2 x2 # t = 2 x1 K 3 x2 . Use the eigendata you have been given to write down the general solution to this system of DE's. (4 points) x1 t x2 t = c1 eKt 1 1 C c2 eK5 t 1 K1 . 2) Consider the following system of second order differential equations, which could arise from an undamped configuration of two masses and three springs connected in parallel, as we've considered this week. x1 ## t =K3 x1 C 2 x2 x2 ## t = 2 x1 K 3 x2 . a) Use the eigendata you have been given to write down the general solution to this system of DE's. (4 points) l =K1 0 w = 1; l =K5 0 w = 5 so, x1 t x2 t = c1 cos t C c2 sin t 1 1 C c3 cos 5 t C c4 sin 5t 1 K1 . b) Solve the initial value problem for this system of DE's, with initial data x1 0 = 2, x1 # 0 = 0, x2 0 = 2, x2 # 0 = 0 . (2 points) x1 0 = 2 = c1 C c3 x2 0 = 2 = c1 K c3 x1 # 0 = 0 = c2 C 5 c4 x2 # 0 = 0 = c2 K 5 c4 . So c1 = 2, c3 = 0, c2 = 0, c4 = 0 .