How would you find the inverse Laplace transformation of $\frac{3s+4}{{s}^{2}-16}$ when s>4?

Jacoby Erickson
2022-10-15
Answered

How would you find the inverse Laplace transformation of $\frac{3s+4}{{s}^{2}-16}$ when s>4?

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Jean Deleon

Answered 2022-10-16
Author has **14** answers

Hint:

Write the partial fraction fraction expansion as:

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{3s+4}{{s}^{2}-16}=\frac{1}{s+4}+\frac{2}{s-4}}\end{array}$$

Now, take the inverse Laplace of each of the terms on the right-hand-side (RHS) of (1).

We have for s>a:

$${\mathcal{L}}^{-1}\left(\frac{1}{s-a}\right)={e}^{at}$$

Write the partial fraction fraction expansion as:

$$\begin{array}{}\text{(1)}& {\displaystyle \frac{3s+4}{{s}^{2}-16}=\frac{1}{s+4}+\frac{2}{s-4}}\end{array}$$

Now, take the inverse Laplace of each of the terms on the right-hand-side (RHS) of (1).

We have for s>a:

$${\mathcal{L}}^{-1}\left(\frac{1}{s-a}\right)={e}^{at}$$

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After looking at the laplace transformations the closest I've found is:

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I've been working out a solution for having three variables, but I cant seem to get the correct solution. Is there an identity for this type of transformation?

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