You have not seen these examples yet but will this week.

First of all, video is coming. Watch, e.g. minutes 2.00 – 11.55 and 25.45 – 27.35 (there is more coming).

Let me talk through this specific example.

O.K. When you look at you think that it looks like

.

Except it is ‘shifted’, the is shifted to . So is more like:

.

See this on ‘top’? You need this. Change the to achieve this:

.

Of course this changes . To ‘balance the books’ you must also add one:

Now this can be split up:

.

This first term is a ‘shifted’ . We can deal with this later.

We have a problem with

.

To be a shifted it must be of the form:

.

By looking at the ‘bottom’ , we know that but we have instead one on top. We want a two… so put a two there:

.

Of course this is not correct. We have changed this function… we have multiplied by two… to fix it we must also divide by two or multiply by one half:

.

Now everything is in order:

.

This is a shifted plus half a shifted .

The First Shift Theorem says that:

.

In English, for shifted functions, bring them back as normal (), and then multiply by an exponential shift (). Here and so the exponential shift is . This is the usual “if come back as we see with the inverse Laplace Transform of .

So is a shifted plus half a shifted so its inverse Laplace Transform is:

.

Regards,

J.P.