Differential Equations - If you could please include all the necessary information that will help in understanding

Differential Equations - If you could please include all the necessary information that will help in understanding

why the answer is the case, including theorems used, general form of equations and the process. Thank you!

Consider the forced damped mechanical system y"+8y’+32y:17166*4t, (1) where y(t) is the displacement at time t and the right hand side corresponds to a non-oscillatory force that is gradually applied. (a) Find the general solution of the associated homogeneous equation3;" + Sy’ + 323; = 0. Note: Use "A" and "B" as your arbitrary constants. yhlt) : (b) Use the method of undetermined coefficients to find a particular solution of (1). 31pm : (c) Hence find the solution of (1) subject to the initial conditions y(0) : 0 and y’(0) : 0.W) : (d) As time t —> oo, determine the behaviour of the forcing function F(t) : 1 — 16 tr“ and your solution y(t) in part (c) above. F(t) —> W) H