$An input x (t) = e^{- 2 t} u (t) is applied to an LTI system whose impulse response is h (t) = t^{2} e^{- t} u (t) . If the output is in the form y (t) = [P e^{- t} + Q t e^{- t} + R t^{2} e^{- t} + S e^{- 2 t}] u (t), then the value of P+Q+R+ S is$ ________.
-1

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Solution

The correct option is A -1
$X (s) = \frac{1}{s + 2}$

$H (s) = \frac{2}{(s + 1)^{3}}$

$Y (s) = X (s) . H (s) = \frac{2}{(s + 1)^{3} (s + 2)}$

$= \frac{A}{s + 1} + \frac{B}{(s + 1)^{2}} + \frac{C}{(s + 1)^{3}} + \frac{D}{s + 2}$

$C = \frac{2}{s + 2} |_{s = - 1} = \frac{2}{2 - 1} = 2$

$B = \frac{d}{d s} [\frac{2}{s + 2}] |_{s = - 1} = \frac{2}{2 - 1} = - 2$

$A = \frac{1}{2!} \frac{d^{2}}{d s^{2}} [\frac{2}{(s + 2)}] |_{s = - 1} = 2$

$Y (s) = \frac{2}{s + 1} - \frac{2}{(s + 1)^{2}} + \frac{2}{(s + 1)^{3}} + \frac{- 2}{(s + 2)}$

$Y (t) = [2 e^{- t} - 2 t e^{- t} + t^{2} e^{- t} - 2 e^{- 2 t}] u (t)$

$= [P e^{- t} + Q t e^{- t} + R t^{2} e^{- t} + S e^{- 2 t}] u (t)$

$\Rightarrow P = 2, Q = - 2, R = 1, S = - 2$

$\Rightarrow P + Q + R + S = 2 - 2 + 1 - 2 = - 1$

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