Finding the Inverse Laplace of X(s)

[Saladsamurai]

Homework Statement

Find the inverse Laplace of:

$$X(s) = \frac{3}{s^2 - 6}$$

I am kind of stuck on this one. I am pretty sure this is not sinusoidal. Can I even use partial fractions on this?

Just a hint here

 

[vela]

It's probably in a table. Or consider the bottom a difference of squares: $$s^2-(\sqrt{6})^2$$ and use partial fractions.

 

[Saladsamurai]

Interesting.

$$X(s) =\frac{3}{s^2 - 6}= \frac{3}{(s+\sqrt{6})(s- \sqrt{6})}= \frac{a}{s + \sqrt6} + \frac{b}{s - \sqrt6}$$

$$\Rightarrow a = \left( \begin{matrix}\frac{3}{(s+\sqrt{6})(s- \sqrt{6})}*({s + \sqrt6})\end{matrix} \right)_ {s\rightarrow -\sqrt6}=\frac{-3}{2\sqrt6}$$

and

$$\Rightarrow b = \left( \begin{matrix}\frac{3}{(s+\sqrt{6})(s- \sqrt{6})}*({s - \sqrt6})\end{matrix} \right)_ {s\rightarrow +\sqrt6}=\frac{3}{2\sqrt6}$$

 

[rock.freak667]

If you wanted you could just use

$$L(sinh(kt))= \frac{k}{s^2-k^2}$$


But partial fractions work just as well.