# Inverse Laplace transform

## Homework Statement

Determine the inverse Laplace transform

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

## 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.

gneill
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?

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

gneill
Mentor
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
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.

Chestermiller
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?

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.

gneill
Mentor
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.

If you go to the Wikipedia page on Laplace Transforms, you will find a match in their table Mark44
Mentor

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

gneill
Mentor
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
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?