Below are four graphs of the position function y(t) over time for harmonic oscil

Question

Below are four graphs of the position function y(t) over time for harmonic oscillators with initial conditions y(0) = 1, y'(0) = 0. Of the eight given equations, choose the four equations that match the graphs, and briefly justify (1-2 phrases) in the space under each graph. (Remark: You should not need to solve the eight equations.) (i) y" + y = 0 (ii) y" + 2y' + y = cos (2t) (iii) y" + 2y' + y = e^2t (iv) 2y" - y'+ 10y = 0 (v) 2y" + y' +10y = 0 (vi) y" + 9y = cos (3t) (vii) y" + 9y = e^-3t (viii) y" + 11y = cos (3t)

Explanation / Answer

(a) The graph matches with the differential equation (v) 2y"+y'+10y=0

It seems like the case of underdamped system.

(b)The graph matches with the differential equation (vii) y"+9y=e^(-3t)

As the time goes on increasing the effect of e^(-3t) keeps on decreasing leading the solution influenced by only sine and cosine terms.

(c)The graph matches with the differential equation (i) y"+y=0

It is our undamped, unforced harmonic oscillator, This has pure periodic solutions. Cosx may be the function.

(d)The graph matches with the differential equation (vii) y"+11y=cos3t

11 is close to 3, Beat will be present.

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