Transforming Inverse Laplace Equations with a Shifting Theorem

[aaronfue]

Homework Statement

L^-1{$\frac{s}{s^2+4s+5}$}

Homework Equations

$\frac{s-a}{(s-a)^2+k^2}$

$\frac{k}{(s-a)^2+k^2}$

The Attempt at a Solution

I completed the square for the denominator and got:

L^-1{$\frac{s}{(s+2)^2+1}$}
(a= -2, k=1)

But how do I get rid of the s in the numerator? Or do I have to break this up into separate functions?

 

[milesyoung]

Say we have:
$$F(s) = \frac{1}{(s+2)^2 + 1}$$
so you need to find $\mathcal{L}^{-1}\left\{s F(s)\right\}$. Have you seen something like that in your transform tables?

 

[HallsofIvy]

The Laplace transform of cos(t} is $$\frac{s}{s^2+ 1}$$. You can find that in any table of Laplace transforms.

 

[LCKurtz]

Also you could write$$\frac s {(s+2)^2+1}=
\frac {(s+2)}{(s+2)^2+1}+\frac{-2}{(s+2)^2+1}$$and use the shifting theorem.