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**Magical** the behavoiur of the solution of d^2x/dt^2+adx/dt+4x

depends on the value of a

Note a=alpha

a)find the ranges of a so that the solution displays

i)overdamping

ii)critical damping

iii)underdamping(decaying oscilations)

iv)growing oscilations

v)constant oscilations

b)For each of these cases, choose value of a within your stated

ranges, and find the general solution, displaying your solution

pictorially to show the behaviour of the solution

c) By carefully considering all the possible values of a identify the ranes which do not lead to the behavouirs listed in a, and describe the typical behaviour in this case