標籤彙整： Laplace transform

【微分方程導論】筆記10

2019-11-27 Sin-iu 發表迴響

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

Example 1

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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微分方程

Properties of the Laplace Transform

2019-11-27 Sin-iu 發表迴響

繼續閱讀 Properties of the Laplace Transform →

微分方程

【微分方程導論】筆記9

2019-11-21 Sin-iu 發表迴響

Theorem 1

(the Laplace transform of the first derivative of $\textit{\textbf{f}}$ )
Suppose:
(1) $f$ is continuous on some interval and of exponential order $\alpha$ ; and
(2) $f'$ is piecewise continuous on some interval.
Then $\mathcal{L}\{f'(t)\}$ exists for $s > \alpha$ and $\displaystyle \mathcal{L}{f'(t)} = s\mathcal{L}{f(t)} - f(0).$

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