Input $x (t)$ and output $y (t)$ of an LTI system are related by the differential equation $y^{''} (t) - y^{'} (t) - 6 y (t) = x (t)$ . If the system is neither causal nor stable, the impulse response h(t) of the system is

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Solution

$y^{''} (t) - y^{'} (t) - 6 y (t) = x (t)$

$H (s) = \frac{Y (s)}{X (s)} = \frac{1}{s^{2} - s - 6} = \frac{1}{(s - 3) (s + 2)}$

$H (s) = \frac{(\frac{1}{5})}{s - 3} + \frac{(- \frac{1}{5})}{s + 2}$

if the system is neither causal nor stable, then

$h (t) = - \frac{1}{5} e^{3 t} u (- t) + \frac{1}{5} e^{- 2 t} u (- t)$

$R O C : σ < - 2.$

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Similar questions

y (t) = \int_{- \infty}^{t - 10} e^{- (t - 5 - τ)} x (τ) d (τ)

y (t) |_{t = 5}

h (t) = e^{- 5 t} u (t)

y (t) = e^{- 3 t} u (t) - e^{- 5 t} u (t)

τ

τ

\frac{d y (t)}{d t} + \frac{1}{6} y (t) = 3 x (t)

x (t) = 3 e^{- t / 3} u (t)

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