Ordinary vs Partial DE

All right, welcome everybody to the first segment of the Differential Equation course. Before actually we go ahead and solve any differential equation, I want to talk about the basic definitions and the terminology so everybody is on the same page, okay? Obviously, this part you may understand that the if I have a differential equation, as the names recommends right over here, right, it says "differential," so I'm kind of going ahead and solving various equations that contain derivatives. So basically, that's the whole thing about it. There’s a differential equation is, I'm going ahead and then solving this this particular equation. But first we have to talk about some basic definitions. Well, why don't we go ahead and write an official definition of differential equation? So let's go ahead and write it. Definition: An equation containing derivatives of one or more (I will talk about this one or more) dependent variables (I'll talk about the dependent variables too) with respect to one or more (so we see metric) independent variables. So sometimes I'm not 100 percent sure everybody knows the difference between those two. This has nothing to do with Differential Equation course, but if I have like y of x, it's a function of f of x, right? You know, very very common thing that we encounter. So this is called the independent variable, okay, and this is called the y, basically it's called the dependent variable. Now, why do I call this dependent variable? Well, because look at it, this y is a function of x, right? So it depends what my value of x is depending on how this f of x is defined, if I am put x is equal to 1 versus x is equal to 5, my value will depend on that number to that I insert. Okay, now what I want to do is I want to go ahead and look at the different types of differential equations. I will classify them by different orders, types, etc. But first let's start with the type. So let's write over here: Classification of DE, so you kind of get the idea that from this point, I'm gonna keep calling it DE as opposed to writing every single time "differential equation", by type. That's the first thing I want to talk about. You're kind of familiar with it, there's actually nothing new in here, okay? The first is called the ODE, so it's the Ordinary DE, and I also have something called PDE, so this is the Partial Differential Equation, okay? So the way that I differentiate these two is looking at this independent variable. The number of independent variables will determine that, okay? If I have like a table over here, number of independent variables over here is going to be one. I will have one single independent variable for to call it to be an ODE. And then, if I have two or more independent variables, then I will call this the Partial Differential Equations, okay? In nature, all of it has applications. I can have ODEs, I can have PDEs. And during this course, I will solve both of them, okay? One thing I want to highlight before giving some examples is, I only clarify by independent variables. And that's right over here, right? So it's the x, if you go with the conventional variable. So this x will be the independent variable, okay? But I don't specify anything with the dependent variable. The dependent can variable can be multiple, and I still call that an ODE. Let's go ahead and give some examples from ODEs. Let's start easy: dy dx minus 2y is equal to e to the power of 2x. So don't get confused. I have a y over here, that's a y over here. So what is the independent variable over here? I'm going to look at this. This is the x, okay? So this can have y over here; that's not gonna be a problem for ODE, okay? So, as I have one independent variable, that I have I have over here, I will go ahead and call this an ODE. Okay? Or I can do a little bit more fancier than that. x cube, as an example, plus, let's say, 2 times dy dx. You can make it as complicated as you want, but let's say minus y, 6y is equal to 0. Okay, so we will solve this, these two equations, in the upcoming segments. But what do I do over here? Okay, this is a bit more trickier now. I have two of these differential equation signs or dy dx type of expression. But look, both of them, if this is x, this is also x. So I can still call this an ODE. Okay, and if this was, for instance, t, time instead of this x, if I have time, then that will be a PDE. Okay, so this is still an ODE. Okay, let me go one more step higher than this. Let's call this, uh, I kind of gave a hint about it when I was talking about it, but this is sometimes throws students off, and I did it on purpose to kind of confuse you a bit. This is an ODE or PDE? You see, I have y and x, but now there's a t. So sometimes students call this a PDE. That's not quite right, because you're not really following the rules. When I go up over here, you see that I look at the independent variable, and okay, I don't panic. I look at the independent variable. I have a t over here. Huh. I have the exact same one, t. So I have only a single independent variable. So this is called an ODE. But do I have one dependent variable now? I have two. y is a dependent variable. x is a dependent variable. That's going to be quite all right. Let's give examples from the PDE side, so we're on the same page. And okay, so I'm going to write an equation that I'm very familiar with because I use it in my research field. So what is this? If I have a one-dimensional rod, okay, I have a single rod, right, circular cross-section, full of, let's say, some metal, right? And this is actually called the diffusivity. This is beyond the scope of this, you know, ODE versus PDE conversation, but I just want to highlight what I'm doing over here. So look over here. This is u, right? And this is u. So how many independent variable, dependent variables do I have? I have one. What about the independent variables? I have x and t. For this reason, I'm going to call this a PDE. And I want to highlight that this is called the heat equation. And I will go ahead and give another example, but we will actually solve these equations that I'm illustrating towards the end of this course. But here's what looks. x square plus del square u dely square is equal to del square u delt square. So this is the wave equation. The c is called the speed of sound, right? So but look here, I have u, u, u. Okay, so the number of dependent variables is one, but when I look at the independent variables, that’s x, y, and t. So this is also a PDE, and this is called the wave equation. I think you get the point of what I'm trying to establish over here, okay? So I can give much more examples from real life as well, but I'm going to stop over here. Okay? Okay, do you want me to give you a really easy way to differentiate those two? Maybe you've already seen it. Do you see there's a difference over here? This is a regular derivative sign. This is a partial. What's the first word? Partial differential equation versus ordinary or regular. You see? So the symbols are different. Okay, and I want to highlight for a second over here what are the differences between those two. From calculus you know this, or you should know this. So basically, if I have a, this is what d y x is. This is the regular derivative. Okay, on the other hand, if you remember the partial, the partial means it's a function of multiple independent variables. It can be x. Well, obviously, it needs to be x. It can be z, it can be time, whatever. I can have multiple independent variables. But the point I'm making over here is I have x over here, and this is the only independent variable over here. This is the independent variable. You can see there are multiple independent variables, plural. So that's how you differentiate those two; ODE and PDE. Thank you for watching the segment.