00:01
In this problem, we want to find the inverse laplace transform of the following function.
00:07
F of s equal to minus the exponential minus square s times s plus 7, divided by s squared plus 64.
00:20
We could certainly calculate the laplace transform from first principles via the integral, but while we reinvent the wheel, we could utilize inverse laplace transformation tables.
00:32
Such tables exist similar as how we use integration tables for integrals, or derivative tables for derivative calculations.
00:45
The goal here is to try to see what kind of shape of function we have and relate this to our laplace transformation tables.
00:54
Of use to us is the following three inverse laplace transformations.
01:02
When you're dealing with an exponential in front of a function, what we have is in fact the inverse laplace transformation of the heaviside function times some function.
01:17
For example, the laplace transform applied to the heaviside function h of t minus c, we call this time shifting, times some function f of t minus c, is equal to the exponential of minus cs times fs, where f is the laplace transformation of small f, where f of t, its laplace transform is equal to f of s.
02:19
Once we do that, we're going to want to compute the laplace transformation of s plus 7 over s squared plus 64.
02:26
To compute this, let's recall the laplace transform of the sine of at and the laplace transform of the cos of at, which respectively yield a over s squared plus a squared and s over s squared plus a squared.
03:06
We're going to want to manipulate our function here to look more like these two inverse laplace transformations.
03:16
Let's start from the beginning, keep these laplace transformations in mind, and see how we can utilize them.
03:29
We want to compute the inverse laplace transform of minus e to the minus 4s times s plus 7 over s squared plus 64...