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Consider the initial value problem y?? + ?y? + y = ?(t ?

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Consider the initial value problem y?? + ?y? + y = ?(t ? 1), y(0) = 0, y? (0) = 0, where ? is the damping coefficient (o

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Consider the initial value problem y?? + ?y? + y = ?(t ? 1), y(0) = 0, y? (0) = 0, where ? is the damping coefficient (or resistance). (a) Let ? = 1 2 . Find the solution of the initial value problem and plot its graph. (b) Find the time t1 at which the solution attains its maximum value. Also find the maximum value y1 of the solution. (c) Let ? = 1 4 and repeat parts (a) and (b). (d) Determine how t1 and y1 vary as ? decreases. What are the values of t1 and y1 when ? = 0?

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