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A system is described by the differential equation d²y/dt² - 5 dy/dt = δ(t) * x(t). Let x(t) be a rectangular pulse given by x(t) = { 1, 0 ≤ t ≤ 2 { 0, otherwise Assuming that y(0) = 0 and dy/dt = 0 at t = 0, the Laplace transform of y(t) is

1. e^(-2s) / [s(s-2)(s-3)]
2. [1 - e^(-2s)] / [s(s-2)(s-3)]
3. e^(-2s) / [(s-2)(s-3)]
4. [1 - e^(-2s)] / [(s-2)(s-3)]

Correct answer: [1 - e^(-2s)] / [s(s-2)(s-3)]

Solution

The correct option represents the Laplace transform of the system's response to the input rectangular pulse, accounting for the initial conditions and the impulse response. The term [1 - e^(-2s)] captures the effect of the pulse from t=0 to t=2, while the denominator s(s-2)(s-3) reflects the system's dynamics as described by the differential equation.

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