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Properties of the Laplace transform Video transcript I'll now introduce you to the concept of the Laplace Transform.

And this is truly one of the most useful Laplace tips that you'll learn, not just in differential equations, but really in mathematics. And especially if you're going to go into engineering, you'll find that the Laplace Transform, besides helping you solve differential equations, also helps you transform functions or waveforms from the time domain to this frequency domain, and study and understand a whole set of phenomena.

But I won't get into all of that yet. Now I'll just teach you what it is. I'll teach you what it is, make you comfortable with the mathematics of it and then in a couple of videos from now, I'll actually show you how it is useful to use it to solve differential equations.

We'll actually solve some of the differential equations we did before, using the previous methods. But we'll keep doing it, and we'll solve more and more difficult problems. So what is the Laplace Transform? That Laplace tips be before many of your times, but I grew up on that.

Actually, I think it was even reruns when I was a kid. So Laplace Transform of some function. And here, the convention, instead of saying f of x, people say f of t.

And the reason is because in a lot of the differential equations or a lot of engineering you actually are converting from a function of time to a function of frequency. And don't worry about that right now.

If it confuses you. But the Laplace Transform of a function of t. It transforms that function into some other function of s. Well actually, let me just do some mathematical notation that probably won't mean much to you. So what does it transform?

Well, the way I think of it is it's kind of a function of functions. A function will take you from one set of-- well, in what we've been dealing with-- one set of numbers to another set of numbers. A transform will take you from one set of functions to another set of functions.

So let me just define this. The Laplace Transform for our purposes is defined as the improper integral. I know I haven't actually done improper integrals just yet, but I'll explain them in a few seconds.

The improper integral from 0 to infinity of e to the minus st times f of t-- so whatever's between the Laplace Transform brackets-- dt. Now that might seem very daunting to you and very confusing, but I'll now do a couple of examples.

Well let's say that f of t is equal to 1. So what is the Laplace Transform of 1? So if f of t is equal to it's just a constant function of time-- well actually, let me just substitute exactly the way I wrote it here.

So that's the improper integral from 0 to infinity of e to the minus st times 1 here. I don't have to rewrite it here, but there's a times 1dt.

And I know that infinity is probably bugging you right now, but we'll deal with that shortly. Actually, let's deal with that right now.

This is the same thing as the limit. And let's say as A approaches infinity of the integral from 0 to Ae to the minus st. Just so you feel a little bit more comfortable with it, you might have guessed that this is the same thing.

Because obviously you can't evaluate infinity, but you could take the limit as something approaches infinity.Enter all or part of a model number and / or product description in the PRODUCT SEARCH box.

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Laplace transform to solve a differential equation Video transcript Let's try to fill in our Laplace transform table a little bit more. And a good place to start is just to write our definition of the Laplace transform. |

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The coefficient of permeability of a soil is a measure of the conductance i. The value of the coefficient of permeability k depends on the average size of the pores and is related to the distribution of particle sizesparticle shape and soil structure. |

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