Beats and resonance suppose that a mass-spring system can be modelled by the differential equation y" +81 y = cos(81) along with the initial conditions y(0) = 0 and y'(0) = 0. (a) describe in physical terms the mass-spring system modelled by this equation. (b) solve this initial value problem, and plot the solution over a meaningful range of t-values. (c) describe the motion of the mass. in particular discuss features of the oscillations such as the max- imum amplitude and period. (you can estimate these from the graph.) (d) what happens to the motion of the mass (for example, changes in amplitude or frequency) if cos(81) is replaced by cos(at) where a is allowed to approach 9? your answer should describe the behaviour in words, but inlude calculations and graphs to support your answer. suggestions: you already have the solution for a = 8. solve the initial value problem for three more values of a close to 9. include a = 9 too. or, you could also solve the differential equation more generally in terms of a, and then substitute values for a. note, though, this analytical solution fails when a = 9 and it takes perseverance to solve this case by hand. desmos or maple will be useful here!