Question 4.2.2: Evaluate L^-1 {-2s+6/s²+4}....
Evaluate \mathscr{L}^{-1}\left\{\frac{-2 s+6}{s^{2}+4}\right\}.
Step-by-Step
We first rewrite the given function of s as two expressions by means of termwise division and then use (1):
\mathscr{L}^{-1}\{\alpha F(s)+\beta G(s)\}=\alpha \mathscr{L}^{-1}\{F(s)\}+\beta \mathscr{L}^{-1}\{G(s)\}, (1)
\begin{array}{c} &\text { termwise division} &\text {linearity and fixing up constants }\\ & \qquad \ \ \downarrow &\downarrow\qquad \\ \mathscr{L}^{-1}\left\{\frac{-2 s+6}{s^{2}+4}\right\} &= \mathscr{L}^{-1}\left\{\frac{-2 s}{s^{2}+4}+\frac{6}{s^{2}+4}\right\}&=-2 \mathscr{L}^{-1}\left\{\frac{s}{s^{2}+4}\right\}+\frac{6}{2} \mathscr{L}^{-1}\left\{\frac{2}{s^{2}+4}\right\} \text { (2) } \qquad \quad \\ &=-2 \cos 2 t+3 \sin 2 t .& \leftarrow \text { parts (e) and (d) of Theorem 4.2.1 with } k=2 \quad \end{array}
Related Answered Questions
Question: 4.2.5
Verified Answer:
Proceeding as in Example 4, we transform the DE by...
Question: 4.2.3
Verified Answer:
There exist unique constants A, B, ...
Question: 4.2.4
Verified Answer:
We first take the transform of each member of the ...
Question: 4.1.6
Verified Answer:
This piecewise-continuous function appears in FIGU...
Question: 4.1.4
Verified Answer:
From Definition 4.1.1 and integration by parts we ...
Question: 4.2.1
Verified Answer:
(a) To match the form given in part (b) of Theorem...
Question: 4.1.3
Verified Answer:
In each case we use Definition 4.1.1.
(a) \...
Question: 4.6.2
Verified Answer:
We must solve
\begin{array}{r} \frac{d i_{...
Question: 4.5.1
Verified Answer:
(a) From (3)
\mathscr{L}\left\{\delta\left(...
Question: 4.4.8
Verified Answer:
Using the result in (15)
\begin{aligned}&am...