Problem 예제.

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$F(s)= \dfrac{2s+3}{s^2+3s+2}$ 의 시간 함수 $f(t)$ 는?

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$f(t) = e^{-t}+2e^{-2t}$

$f(t) = e^{-t}-2e^{-2t}$

$f(t) = e^{-t}+e^{-2t}$

$f(t) = e^{-t}-e^{-2t}$

Solution

$\dfrac{2s+3}{s^2+3s+2} = \dfrac{2s+3}{(s+1)(s+2)} = \dfrac{k_1}{s+1} + \dfrac{k_2}{s+2}$

$k_1 = \lim_{s \to -1} (s+1) \dfrac{2s+3}{(s+1)(s+2)} = 1$

$k_2 = \lim_{s \to -2} (s+2) \dfrac{2s+3}{(s+1)(s+2)} = 1$

$\mathcal{L}^{-1}\left[ \dfrac{1}{s+1} + \dfrac{2}{s+2} \right] = e^{-t}+e^{-2t}$

Correct answer: 3