# Inverse Laplace transform

1. Jan 21, 2017

### casper12

1. The problem statement, all variables and given/known data
Determine the inverse Laplace transform

2. Relevant equations
3s+9/(s+3)^2+7

3. The attempt at a solution
Converted to 3s+9/s^2+6s+16 to try and use the partial fractions method but getting nowhere.

I'm Not sure if Iam making the question more difficult, can't seem to put the division side of the equestion into brackets.

Attempt was (s-2)(s+8) but that results in s^2+6s-16 resulting in an incorrect sign. Can't seem to change the signs to make it correct.

Not sure if partial fractions method is the best method or if I'm making it more difficult than it is.

2. Jan 21, 2017

### Staff: Mentor

Please use more parentheses (brackets) to make the order of operations in your expressions clear. Is that:

$3s + \frac{9}{(s+3)^2} + 7$

or

$3s + \frac{9}{(s+3)^2 + 7}$

or

$\frac{3s + 9}{(s+3)^2} + 7$

or

$\frac{3s + 9}{(s+3)^2 + 7}$

or something else again?

3. Jan 21, 2017

### casper12

Sorry not posted many threads and not familiar with the way you have written the equation so well. The last equation.

3s + 9.
(s+3)^2+7

4. Jan 21, 2017

### Staff: Mentor

Okay. The syntax used to display equations is LaTeX syntax. You can find some information on LaTeX by following the link at the bottom left of the Reply edit window (? LaTeX / BBcode Guides).

Are you expected to derive the inverse Laplace transform for this expression via the definition, or merely recognize its form and look it up in a table of inverse transforms? If you need to derive it, I think you would do well to take this to the Calculus & Beyond homework forum as it will involve integrating a complex function and taking limits. I suspect that the Cauchy Residue Theorem will come into play for finding the integral.

5. Jan 21, 2017

### Staff: Mentor

I agree with gneill. If you had $\frac{s}{s^2+7}$, would you be able to find its inverse in a tables of laplace transforms?

6. Jan 21, 2017

### casper12

The question is asking for the inverse Laplace transform of the equation using the table of Laplace transforms, all the ones completed in the workbook at not of this format so difficult to find a similar method to calculate. I have posted the question where you recommended, thank you for your help.

7. Jan 21, 2017

### Staff: Mentor

If you go to the Wikipedia page on Laplace Transforms, you will find a match in their table

8. Jan 21, 2017

### Staff: Mentor

Last edited: Jan 21, 2017
9. Jan 21, 2017

### Staff: Mentor

10. Jan 21, 2017

### casper12

Thanks for the Wikipedia info just seen the table which includes a lot more Laplace transforms than the info provided in the workbook. Looking at this it looks something like
3 x e-3t cos wt
any further advice would be much appreciated as I believe I may have wasted a lot of time on this question.

Thanks

11. Jan 21, 2017

### Staff: Mentor

Yes, it's an exponentially decaying cosine. The Wikipedia tables can be quite handy.

Is there anything further you need to do with this question?