# Math Help - My curiosity needs satisfying

1. ## My curiosity needs satisfying

I had a maths exam last Monday and the only question I couldn't do is below. Can anyone shed any light?

Laplace transform the following:

1/(s^2+s+5/4)

2. Don't you mean the inverse LaPlace?.

In that event, $\frac{sin(t)}{\sqrt{e^{t}}}$

3. Originally Posted by boab
I had a maths exam last Monday and the only question I couldn't do is below. Can anyone shed any light?

Laplace transform the following:

1/(s^2+s+5/4)

Inverse Laplace:

$\mathcal{L} ^{-1} \left\{\frac{1}{s^2+s+\frac{5}{4}}\right\}$
$\implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s^2+s+\frac{1}{4}\right)+\fr ac{5}{4}-\frac{1}{4}}\right\}$
$\implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s+\frac{1}{2}\right)^2+1}\ri ght\}$

Apply the first translation theorem:

$\implies \mathcal{L}^{-1} \left\{\frac{1}{\left(s+\frac{1}{2}\right)^2+1}\ri ght\}$

$\implies f(t)=e^{-\frac{1}{2}t}\sin\left(t\right)$

$\implies \color{red}\boxed{f(t)=\frac{\sin\left(t\right)}{\ sqrt{e^t}}}$