Question

Criminal masterminds assume that the sways of the Tower are governed by the ODE:

x¨ +2α x˙ +(α²+ω² )x=F(t)
where α and ω are constants and F(t) denotes the forcing they will apply. The most brilliant part of the plan is to apply the forcing over a carefully chosen interval 0 < t < T, such that sways completely disappear once the forcing ends (i.e. for t > T), creating the perfect crime. As a proof of concept, the criminals solve the ODE by setting F = 1 for 0 < t < T, and then F(t) = 0 for t > T, taking ω = 1 and α = x(0) = x˙ (0) = 0. Using Laplace transforms, step functions and a shifting theorem, follow their steps to find a solution for x(t). Then consider t > T and show that x(t) = 0 for t > T if T is suitably selected. ​

Answer 1

To solve the given ODE and find a solution for x(t), the criminals utilize Laplace transforms, step functions, and the shifting theorem. They derive the solution x(t) = Acos(ωt) for 0 < t < T and demonstrate that x(t) = 0 for t > T.

Step-by-step explanation:

The given ODE, x¨ +2α x˙ +(α²+ω² )x=F(t), represents the sways of the Tower governed by certain parameters. To solve the ODE and find a solution for x(t), the criminals set F = 1 for 0 < t < T and then F(t) = 0 for t > T. Using Laplace transforms, step functions, and the shifting theorem, the solution for x(t) can be derived as x(t) = Acos(ωt) for 0 < t < T and x(t) = 0 for t > T.