# Having difficulty finding the inverse laplace transform

Having difficulty finding the inverse laplace transform!!

Hello everyone, I am really stuck on finding the inverse Laplace transform for this:

$$f(s)=\frac{5se^{-3s} - e^{-3s}}{s^{2}-4s+17}$$

Heres my reasoning: I feel that I should rewrite the denominator in some kind of form such as (s-2)^2 + 13, and note the similarity with some of the problems ive been doing before, however its that 13 that is bothering me! Its not something you can take the squareroot of--- and also in addtion, I tried factoring out e^-3s on top and splitting this into two equations, but its this denominator that I absolutely despise.

Any help with finding the right method would be greatly appreciated thank you!!!

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arildno
Homework Helper
Gold Member
Dearly Missed
I've forgotten just about every transform formula I knew (and I'm not in the mood rederiving them), but you may write $$13=(\sqrt{13})^{2}$$

if that helps..

hmm i still am having difficulty---

arildno
Homework Helper
Gold Member
Dearly Missed
$$e^{-3s}=e^{-6}e^{-3(s-2)}$$

Then you would get one expression on the form:
$$k\frac{e^{-3w}}{w^{2}+a^{2}},k=-e^{-6},w=s-2,a=\sqrt{13}$$

Is this a familiar transform in w?

Tom Mattson
Staff Emeritus
Gold Member
The exponentials are due to shifts in the t-domain:

L{f(t-T)}=e-sTF(s).

Just find the inverse transform of the rational functions of s, and then let the t-domain functions be delayed by the appropriate amount, given by the coefficient in the exponents of the exp functions.

interesting let me see if i can get anywhere with that...

Uh oh, you see, I cannot split the denominator into two linear factors like the book seems to be doing with most of the problems---- and because I cant, i dont know how to proceed like the examples do... Im sorry if im a bit slow but we just started Laplace and this is a challenge problem id like to know to prepare.

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Well I cannot solve it but thanks for your help anyway...

Tom Mattson
Staff Emeritus