## Ciel_Phantomhive 3 years ago inverse Laplace Transform F(s)= (s^2+2)/(s^2+2s+5)

1. satellite73

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. satellite73

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

3. Ciel_Phantomhive

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

4. satellite73

that i can do

5. Ciel_Phantomhive

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

6. satellite73

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. Ciel_Phantomhive

what did you divide by?

8. satellite73

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. satellite73

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

10. satellite73

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

11. Ciel_Phantomhive

Oh okay Thanks

12. vf321

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. Ciel_Phantomhive

Oh I have forgotten that, thanks

14. vf321

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

15. satellite73

yeah it is in there, rest looks like greek

16. vf321

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

17. satellite73

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

18. satellite73

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

19. vf321

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!