 Okay, let's start with a brand new section and we're going to do what is called the Laplace transform Laplace transform First of all, let's have a look at what a transform is a transform is Exactly as the word implies we are transforming something if I have a function f of x Equal say for instance x cubed and I take the first derivative of the f of x With respect to x that is going to be 3x cubed Squared I should say 3x squared So whereas before I had a function which was x cubed I now have a new function called 3x squared. I have transformed this function into something else I have transformed f of x and In the transform I Performed was applying the first derivative with respect to x that was a transform we're now going to have a new transform and It is going to be in the form of a very special integral we're going to do a special kind of integration well, it's still normal integration, but we're going to add something to a function and add something to it Let's show how should we do this. Let's do this. We're going to add something to this Function and then we're going to integrate it Are we going to do? This type of integration to it Now suppose here it would be DX Usually we use t in this instance, so we're going to have a function f of t And we're going to take the plus transform of the f of t. It's written like this in L Curse of L And then these curly braces the f of t Close curly braces. It is taking the plus transform or something This was taking the first derivative with respect to x everything we do has its own form of writing So yet we use this notation to say we're transforming this function We're now going to use this notation to say we're transforming this function And how do we do that? What is this part that gets placed in here? Now, of course, you can see that we the eventual integral is going to be a Improper integral from zero to infinity, but what is this that we put in there this thing? This is a very un-mathematical way of doing this Is e to the power negative s t? Such that s is larger than zero We definitely want this to be one over the s of e to the power s t Definitely got to be in the denominator. So we want that s to be at least a larger than zero Just as it stands here Depending on what the answer is and the answer is always going to be some new function Let's call it up a case of f just to distinguish the two and it's also always going to be a function of s If f was just a function of t and I introduce this I'm going to end up with a function in s always going to see that Okay, so let's have the f of t being a constant and that constant was one So if I take the Laplace transform of one Of one What is that going to be? Well, we said first of all we're going to do the improper integral. So that's going from zero to infinity With respect to t we always write these things with respect to t I suppose I could put an x there as long as I then have an x there as well So I'm going to this becomes this e to the power negative s t times the function itself and this instance is just Deat, it's just one. So if s is this constant which is now at least At least larger than Zero What is the integral of this while it's negative one over s? Times e to the power negative s t going from zero to infinity Now, how did we do these initially? How did we do these initially remember when in first year calculus? We used to write something like this as b goes to infinity of the integral of zane from zero to b of e to the power negative s t dt that's how we would have done it So we would have got a function and then we would have seen if there's a limit if there was a limit This was convergent if there's no limit didn't exist This was divergent and we couldn't this is an area under a curve So there wouldn't be an area under the curve as b goes to infinity It wouldn't be a definitive area under the curve. It wouldn't be convergent. Okay So for these remember L'Opital's rule and all the rules we have for these limits And we always just use shorthand instead of going to the b and then putting in b We just just do that but remember what should happen behind the scenes. So what do we have left here? It's negative one over s and What is this? Well, it is One over e to the power s t because there's a negative there and this has got to go from Zero to infinity So I'm gonna have this negative one over s I'm going to have this negative one over s and What are we going to have? We're gonna have one over e to the power s times infinity Minus one over e to the power s times zero just putting those two in One over anything to the power infinity. Well, that's the zero. So I'm gonna have one over negative one over s still That is a zero minus Anything to the s to s times zero is zero e to the power zero is one so it's minus one So that equals one over s As I said to you it will always be a function of s so the Laplace transform of one equals one over s equals one over s and I remind you what the Laplace transform is the Laplace transform of a function of t equals the improper integral going from negative from zero to zero to infinity of e to the power negative s t Times the f of t. Yeah, it was just one Dt so that is the Laplace transform You have to remember that and The definition is what is a transform you transforming a function And in this instance, it is called the Laplace transform. It's difficult to get used to in the beginning To remember how to do all of these eventually it will become second nature and you'll start We'll start to show what it is useful for why we would transform a function As a differential equation and I can tell you now. It's because it changes it into a normal Function that we can just do algebraic manipulation on actually Makes the differential equation easier to solve another form of solving linear differential equations By taking the transform first and Laplace transform That is how to do it That's the equation and here we have our first one of the constant one would be one of s