Vibrations with external force

Problem 5 (Bonus: Vibrations with external force). The harmonic oscillator equation describes Vibrations without any external force applied to the system. We can consider a model in which we apply to the system a periodic external force F (t) = P0 cos (wt). Here 0) is a constant, the frequency of the external force (which is not necessarily the same as the frequency as the system 010 = \\/ k / m). The differential equation we get is my\" + ky 2 F0 c0s(wt). (0.3) Find the general solution to this equation. There are two cases. 1. When 02 7E we, look for a particular solution using the method from Problem 3, by first solving the corresponding complex problem my" + ky = 1300\""t (0.4) and then taking the real part of the complex solution. Look for a solution to (0.4) of the form y(t) = 516\"\" where a is a complex number. 2. When (0 = we the complex equation (0.4) has no solutions of the form y(t) = aeiwt. Instead, look for a complex solution of the form y(t) 2 ate\" for a complex number a. Then take the real part to find a real solution to (0.3). In both cases, sketch the graph of the solution. What is the difference between the behavior of the system in these two cases