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A system is described by the differential equation $\dfrac{\mathrm{d}^{2} y}{\mathrm{d} x} + 5\dfrac{\mathrm{d}y }{\mathrm{d} x} + 6y(t) = x(t).$

Let $x(t)$ be a rectangular pulse given by

$x(t) = \begin{cases} 1&0<t<2 \\ 0&\text{otherwise} \end{cases}$

Assuming that $y(0) = 0$ and $\dfrac{\mathrm{d}y }{\mathrm{d} x} = 0$ at $t=0,$ the Laplace transform of $y(t)$ is

  1. $\frac{e^{-2s}}{s(s+2)(s+3)} \\$
  2. $\frac{1-e^{-2s}}{s(s+2)(s+3)} \\$
  3. $\frac{e^{-2s}}{(s+2)(s+3)} \\$
  4. $\frac{1-e^{-2s}}{s(s+2)(s+3)} $
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