# Math Help - Inverse Laplace transform problem 1

1. ## Inverse Laplace transform problem 1

Hello,

I got the following differential equation :
$E_2'(t)+2{}E_2(t) = E_1(t) + E_1'(t)$

Transforming this into a Laplace equation and computing the transfer function, I get :
$H(s) = {{s+1} \over {s+2}}$

Where I get into trouble is to obtain the inverse transformation. It seems probably silly. I do not have any problem solving more complex equations using partial fractions, but trying, this does not seem to apply here :
${{s+1} \over {s+2}} = {A \over {s+2}}$

What am I missing here?

Thanks,

Jurgen

2. Take into account that:

$\displaystyle\lim_{s \to{+}\infty}{\dfrac{s+1}{s+2}}\neq 0$

Fernando Revilla

3. ... and also that is...

$\displaystyle \frac{s+1}{s+2}= 1-\frac{1}{s+2}$

Kind regards

$\chi$ $\sigma$

4. ... and also we deduce that

$\mathcal{L}^{-1}\left\{{ \dfrac{s+1}{s+2} }\right\}$

does not exist.

Fernando Revilla

5. ## Reply : Inverse Laplace transform problem 1

Why not? I have controlled the answer above yours, which says
$\displaystyle \frac{s+1}{s+2}= 1-\frac{1}{s+2}$

which then translates to

$\delta{(t)}-e^{-2t}$

6. It is important to observe that $H(s)$ is a Transfer function, i.e. the ratio of the L-transforms of the 'output signal' $e_{2} (t)$ and the 'input signal' $e_{1} (t)$ . In this case the $H(s)$ is a [simple] first order 'high pass filter'...

Kind regards

$\chi$ $\sigma$

7. Originally Posted by chthon
$\delta{(t)}-e^{-2t}$
$\dfrac{s+1}{s+2}$