【稼軒詞】上課筆記11/28

浣溪沙 (p.454)
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【微分方程導論】筆記10

Let’s look at some examples to illustrate the s-axis translation theorem (Theorem 3 introduced in the last lecture).

Find the inverse Laplace transform of $\displaystyle \frac{1}{s^2-4s+5}$ .

Solution
Let $\displaystyle F(s) = \frac{1}{s^2+1}$ and $\displaystyle f(t) = \mathcal{L}^{-1}\{F(s)\} = \sin t$ . Then because

$\displaystyle \mathcal{L}^{-1}\left\{\frac{1}{s^2-4s+5}\right\} = \mathcal{L}^{-1}\left\{\frac{1}{(s-2)^2+1}\right\} = \mathcal{L}^{-1}\{F(s-2)\}$ ,

by s-axis translation theorem, $\displaystyle \mathcal{L}^{-1}\{F(s-2)\} = e^{2t}f(t) = e^{2t}\sin t.$ ❏

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