## anonymous 4 years ago inverse Laplace Transform F(s)= (s^2+2)/(s^2+2s+5)

1. anonymous

no idea how to do this, but if you can find an answer it should match up with this http://www.wolframalpha.com/input/?i=inverse+laplace+%28s^2%2B2%29%2F%28s^2%2B2x%2B5%29

2. anonymous

think it is partial fractions and then a formula, but really i should just shut up

3. anonymous

Yeah but before the use of partial expansion you have to to make it a proper fraction.

4. anonymous

that i can do

5. anonymous

Will it be (s^2+2)/(s+1)^2+4?

6. anonymous

divide first $\frac{s^2+2}{s^2+2s+5}=\frac{s^2+2s+5-2s-3}{s^2+2s+5}$ $=1+\frac{-2x-3}{s^2+2x+5}$

7. anonymous

what did you divide by?

8. anonymous

i just arranged it so that the power up top was less than the power in the denominator it is either that, or long division

9. anonymous

it is just an algebra trick so you don't have to divide

10. anonymous

added and subtracted what i needed to match up with the denominator

11. anonymous

Oh okay Thanks

12. anonymous

inverse laplace of 1 is the dirac delta - was that in the answer? Because if it wasn't then you may have added a solution...

13. anonymous

Oh I have forgotten that, thanks

14. anonymous

Ah, yes. I just looked at the Wolfram|Alpha link. It did indeed have the delta function.

15. anonymous

yeah it is in there, rest looks like greek

16. anonymous

Well the rest falls out of all of that fun stuff from DE class that I don't remember...

17. anonymous

i know you use partial fractions, and then tables i think

18. anonymous

but this is when i really should shut up, because i have no idea, although it seems like something i ought to learn

19. anonymous

I'm pretty sure OP is familiar with relations such as $\mathcal{L}^{-1}\{\frac{1}{s^n}\} = \frac{t^{n-1}}{\Gamma (n)}$But yeah the hard part is just the setup, the rest is looking at the table!