Notations

Okay, now that I know the difference between an ODE and a PDE, now I can go ahead and introduce the notations that I will be using throughout this class. Okay, what I mean notation is I keep writing this dy dx. Did you see that? That's actually a specific notation, and that notation is called, let's actually write here, notation. The first one is the one that I personally like to use, and I'll talk about why in a moment. Okay, we know this, I just write it to you. This is dy dx, or I can have, I don't know, d cube y dx cube, or in general terms, I can have dn y dx n. So this is the Leibniz notation. So it's a bit longer than some of the others that I'm going to introduce, but one thing that I want to highlight right off the bat is I clearly see what my dependent variable is, what is my independent variable. Okay, if you are wanting to write this a bit shorter, what you can do is you can write a prime notation. I'm almost certain you've seen this in your undergraduate classes, other undergraduate classes. So it's y prime, y third. We're not gonna write y, you know, like this. We don't do that. Okay, instead we start writing y to the n after, you know, the fourth and above. Okay, so let's rewrite an equation that I had before with Leibniz notation. So let's say this: dx cube plus 2 dy dx minus, I think I called it 6, but it doesn't really matter for our purposes, and I call this an ODE, right? So let's write this in prime notation. So this is gonna be y one two three plus 2 y prime minus 6 y is equal to 0. Much nicer, right? Let's be honest, this is much better than this, correct? What I've experienced, though, in real life is that whenever I have like a shortcut, like it looks really nice, there's usually a disadvantage that is associated with it. And this is one of those cases as well. What is the disadvantage over here? Okay, and what I'm going to do is I'm going to actually go ahead and get rid of this. So now the question is what is the independent variable? What is the dependent variable in here? Can this be dy dt, right? Or is it with respect to the x, right? So this information is lost. If I use prime notation, I don't know which one is right. I really don't know which one is the most appropriate one. Okay, so that is a disadvantage of using the prime notation. Be careful about it. This is the two common notations that we use with ODEs, but there's a third one. Let me write it over here as well. It is the Newton's dot notation. But this when goes one step about the prime notation because, as I said, I'm kind of confused with what is the independent variable in here. The Newton's dot notation is only written with an independent variable of t. So that's the only way that I write this independent variable. Okay, so that’s, so for instance, if I have d square y dt time square, is equal to y double dot, like that. Okay, that's an example. If I have, let’s call, you know, s, but it's always t as notation. And I will also introduce a notation. It's called the subscript notation. But one thing that we need to be careful about it, this is typically used with PDEs, not ODEs, and you'll see why in a minute if I write it. And the good thing about it, it indicates my independent variable. Let's write the PDE. Del square u delx square is equal to del square u dely square minus delu del t, as an example. Okay, why is this a PDE and not an ODE? Well, you can see x, y, t, which u is the dependent variable, right? So this is clearly a PDE. So I can write this this way: u subscript, that's where the name comes from, xx is equal to u yy minus u t. So how many I write over here determines the order of it. The way that I write, you see xx, yy. As I have the first order over here, I just write t. Okay, the good thing about the subscript notation, and I do use this, is that it's very clear what I'm talking about with respect to dependent and independent variable. Okay, so far so good. What we did was we classified the DE, whether it's an ODE or PDE. Then after introduced that, I introduced some notations, how to express the ODE and PDE. Now I'm going to do another classification. But this time around, I'm going to do the classification by order. Okay, classification by order. So basically, this is actually fairly straightforward, except I'm going to give you a challenging question in a minute. But the order of either PDE or ODE is the order of the highest derivative. It is equal to the order of either a PDE or ODE. You will decide that. Okay, so I look at the highest derivative. What is the order of that? That will be the order of the equation itself. Okay, let me give you a fairly straightforward question to you. Well, d square y dx square plus 2x is equal to 5. So what is the order of this? Well, I know you're, I'm hearing almost what you're saying. It's the second order, right? So that's that. You know this is doable. All right, so I'm gonna now try to look at this. How about this, let me write it, then I'll talk. dt power of 5 square plus 6y is equal to 0. I'm making this up. So this is the 10th, right? This is the 5, so the square of it, that will be the 10th order, right? That will be wrong. No, this is the fifth order. I'm looking at this. You know, you need to be careful of where order of the highest derivative, what is the order of the highest derivative? It's the square of it, I get that part, but it's still the fifth order. Okay, so this is called the fifth order ODE. Okay, from the same logic, I can try to fool you one more. Let's do this. dx cube plus 2 times dy dx to the power of 4 is equal to e to the minus x, as an example. Right, so what is the order of it? It is, the fourth order would be the wrong answer. Right, because this is the fourth power of the first order, but rather I have over here a third order. So from this logic, I'm gonna get myself a third order ODE. Very good. So we finish the classification by order, and again I don't see much issues from this classification by order after I explain this way. Okay. So everybody kind of gets it. Next, what I want to do is I'm going to look at several forms of representing DEs, okay? The first one, it's kind of specific to only the first order, but it's called the differential form of a first-order ODE, and it can be written like that: M x comma y dx plus N x comma y dy is equal to zero. Can you write it this way? Absolutely. You see, I can move this dy. I can move this dx down here, you know, I can organize this so I get myself a nice function. So yeah, this is makes sense. And actually, please note that I will use differential form to solve first-order differential equations when time comes. Okay, another form is called the general form of. This time around, I'm going to write this for the n-th order ODE. This can be written for generalized. Okay, so here is that oh, it's going to F x y y prime. What? Well, I just said it. I was going to ask what notation is it. The y prime is what I said, so prime notation. So I go all the way to the power of n is equal to zero. Okay, so can I write an n-th order differential equation this way? Absolutely, and I'll talk about this as well. We do use this. And another form that we use is called the normal form of n-th order ODE. So this is now is going to be written like that. I like to highlight that the right-hand side of this is 0, right? So now I don't have any zero anymore. So this is the n-th order of it, to the power of n, will be a function of x y. Rest of is very similar, except y n minus one. I'd like you to look here and here. Do you see the difference? Okay, so basically what I do between the general and normal form is I take out this term and I represent it this way instead of saying this. Do you see? So it's not that different from one another. General or normal form, we use them all and we can use them interchangeably. We can simply convert that from general form to the normal form as desired. Okay, this may get confusing, so I'm going to solve an easy question to illustrate how to look at, for instance, differential and normal, as an example. Okay, so let me give you a simple function because we're just getting started. I don't want to scare you so you don't leave my course. 5x y prime plus y is equal to x. Obviously, prime notation is what I'm using. The question is, well, let's first start with this: what is the normal form of it, and what is the differential form of it? Okay, that's the question that I'm looking at over here. Okay, first thing I'm going to do is I'm going to insert this this way: 5x dy dx. Leibniz notation. It's much more straightforward for me. Okay, obviously I'm assuming this is, although it's not clear, I'm assuming this is x. Good. But what am I trying to achieve? So let's look at the normal form. Here's what it says. I will have, what is the order of this equation? It's a first order, right? Also, you can get a hint that as I'm asking you differential form, that is only defined for first order. So I have, right? But anyways, so I'm looking over here. So how I want to represent this is like this: dy dx is equal to f of x comma y. I don't have y prime and above. Why? Because y prime is right, actually this, right? So this is how I need to express this. Is this a big deal? I don't think so. Let's do it. dy d x. So what I'm going to do is I'm going to manipulate this. So it's going to look like this: 5x dy dx, and I'm going to move that y to the right-hand side of the equation. So it's going to be x minus y. Let's divide both sides by 5 5x, assuming x is not zero. I'm going to get myself x minus y divided by 5x. Right, so this is pretty much it then. You see the difference? You see? So now this is the normal form. Okay, so I accomplished that. You can see it's not a big deal. Normal form. Let's look at the differential form. But before that, let's look at the definition of it. You can see I need to write this: dx times some function plus dy times some function is equal to zero. Okay, I don't think it's terrible. Let's take a look at it. Um, where should I start? Why don't we start here? Okay, why don't we start here. So you can see over here, if I start from there, so differential form will be 5x times dy is equal to x minus y times dx. Right? I simply move this dx to this side of the equation. And then I need to do one last step, and that is 5x dy. If I move it to the other side, it's going to be a minus sign. But let's write it this way: y minus x. Do you see what I did there? I just swapped this, so I get a positive sign. dx is equal to zero. So if I look at the above equation that I have, this, whatever in front of dy is called the N, right? So this N. And whatever is in front of x is called M. So then from here, you can see that this is going to be my N, this is going to be my M. But that's it. Thank you for watching this video.