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Math 2280-001 Quiz 11 SOLUTIONS April 17, 2015 1a) Use the Laplace transforms to solve the IVP F0 2 x##C w0 x t = cos w0 t m x 0 = x0 x# 0 = v0 (5 points) Solution: The differential equation is an equality for the actual solution x t , so the Laplace transforms of both sides also agree: F0 2 s s2 X s K sx0 K v0 C w0 X S = m s2 C w2 0 2 s2 C w0 = X s X s = F0 m s F0 s x t = 2 2 2 C w0 F0 C s x0 C v0 m s2 C w2 0 C x0 t sin w0 t 2 w0 m s s s 2 2 C w0 C w0 v0 w0 s2 C w2 0 C x0 cos w0 t C v0 w0 . sin w0 t . 1b) What phenomenon do solutions to 1a exhibit? (1 points) Resonance - the first term in the solution above has linearly growing amplitude. 2) Use the table to compute the inverse Laplace transform LK1 F s t for 5 3 4s F s = C C 2 . 2 s sK3 s C 2 s C 10 (4 points) solution: F s = 5 C s 3 sK3 2 C 4 sC1 sC1 2 C9 K 4 3 3 2 sC1 so x t = 5 C 3 t e3 t C 4 eKt cos 3 t K 4 Kt e sin 3 t . 3 C9