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Laplace Transform and the Driven Oscillator

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Please provide help on this assignment with a clear explanation.
1. A mechanical system is described by the differential equation :
y''+(w^2)y=f(t)1, 0 < t < a
f(t) = { y0=y'0=0 0,
otherwise (1/w^2) (1-cos(wt)) t < a
show that : y = { (1/w^2) [(cos(w)(t-a) - cos(wt)] t > a

2. Sketch the motion if T is the period for free vibration of the system and:
a = T/3 , a =( 3/2) T , a = T/10

3. As the convolution integral, consider t < a and t > a separately f(t) = 0 for t > a.

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Solution Summary

The solution shows how to use the convolution integral to solve the harmonic oscillator equation when driven by a time-limited force.

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