Using the Table of Laplace transforms, find the inverse Laplace transforms of each of the follow

Using the Table of Laplace transforms, find the inverse...

Key Concepts

Frequency Shifting Theorem

The frequency shifting theorem, also known as the exponential shift property, states that a shift in the s-domain (i.e., replacing s by s-a) corresponds to multiplying the time-domain function by an exponential term, e^(at). This property is crucial when the Laplace transform exhibits shifted arguments in the denominator.

Standard Laplace Transform Pairs

These are well-known relationships between common functions (such as powers of t or trigonometric functions) and their Laplace transforms. Familiarity with these pairs allows for the immediate identification of functions in the time domain when encountering standard s-domain expressions like 1/(s^n) or s/(s^2 + a^2).

Inverse Laplace Transform

This concept involves converting a function from the Laplace (s) domain back into the time domain. It relies on known transform pairs or analytical methods to identify the original time?domain function that corresponds to a given Laplace domain expression.

Laplace Transform Table

A Laplace transform table is a compiled list of common functions and their Laplace transforms. This resource is essential for quickly identifying standard transform pairs and is often used to immediately determine the inverse transforms of given functions by matching their algebraic forms.

Linearity of the Laplace Transform

The linearity property indicates that the Laplace transform of a sum of functions is the sum of their individual Laplace transforms, and similarly for constant multiples. This property allows one to break down more complicated transforms into simpler parts whose inverses are known.

Transcript

00:01 Well, this problem has to do with using a table to find inverse laplace transforms.

00:09 And so here are the four problems.

00:15 And i have shown on the left a short table of laplace transforms.

00:20 The first two problems a and b are in red and the appropriate applicable transform pairs are shown in red.

00:34 Items one and two.

00:36 And so the first thing you have to do is to decide which transform pairs can be used, and maybe several can be.

00:45 Then you have to decide which one or ones will you use.

00:51 And to do this, part of it is deciding after you look at the table entries, which typically are in terms of generic quantities like a, b, and n, what is a, what is b, what is n in your particular problem.

01:10 And so this is kind of a pattern matching process that goes on there.

01:14 And what other properties might you need? mostly you will need linearity, but you might need some other properties of laplace transforms.

01:26 So if we start with problem a, we have f of s is equal to minus 5 over s minus 2 squared.

01:37 And the question is what is f of x because in your book x is your independent variable here and so we have two applicable properties and property number two is very similar to property number one but it has a equal to zero in it because e to the zero x is a but in any case, in this case, number one, entry number one is the most obviously applicable.

02:20 And so what you need to do is you need to identify what is a and what is n.

02:27 And the denominator is s minus 2 squared, so n is 1, because in the table it's n is the power in the denominator is n plus 1.

02:41 So n is 1, and a is 2.

02:47 And so i have rewritten this with the minus 5, multiplying 1 factorial over s minus 2 to the 1 plus 1.

02:57 And so i'm going to be using linearity here.

03:02 And so if i inverse laplace transform, the minus 5 is a constant, so that comes out.

03:08 With n equal to 1 and a equal to 2, formula transform pair number 1, says that that should be involved an x to the first power and an e to the 2x.

03:25 And x to the first power is just x, so we did a little simplification there.

03:30 So our f of x is minus 5x, e to the 2x.

03:38 Okay, well, part b is f of s is equal to 2, e to the third power, sq.

03:48 Well, e is just a number.

03:49 It's approximately 2 .7 -1828 -1828 -18 -48 -48 -48 -48 -48 -45 -9 -e -45.

03:55 It's a transcendental number, so it continues without repeating.

04:02 But it's just a number.

04:06 And so what is f of x here? well, again, we're going to have to use linearity.

04:12 And the bottom number two is the most applicable, but it's, it's the same as number one just with a equal to zero.

04:24 So 2.

04:25 E to the third is just a constant.

04:28 So i bring that out.

04:30 And my formula has a 2 factorial over s, has an n factorial over s to the n plus 1.

04:37 So if n is 2, that would be 2 factorial over s to the 2 plus 1.

04:43 So i include the 1 over 2 factorial to make that a 1.

04:49 Basically i'm multiplying by 1, 2 factorial over 2 factorial.

04:56 And so i have 2, e to the third power.

05:03 The 1 over 2 factorial puts a 2 in that denominator, and then the inverse laplace transform of 2 factorial over s to the 2 plus 1 gives me my x squared.

05:16 Cancelling 2s, i just get e to the 3rd power x squared.

05:25 So again, there's usually a little.

05:27 Little bit of simplification that goes along in this.

05:33 Using my tables, again, in this case, my c, part c, is f of s is equal to 3 to the s squared plus 9.

05:44 What is f of x? well, my denominator is s squared plus something, a positive something.

05:53 And so that looks like one of these four might be usable...

Please give Ace some feedback