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So as we draw near to the end of this course, I want to do one more topic that really helps transition into course. You may potentially take later, which is differential equations, but then also work on a very interesting problem from physics that uses a lot of information that we have developed in this course. So it's kind of like you can think about it as a end of course project, if you will. But first of all, I want to do just a quick review of differential equations because we talked about them very briefly. But we really know enough now to give some of the preliminary definitions and differential equations. So recall that a differential equation is an equation, of course, that relates a function in its drip tips, now the highest derivative, my highest derivative. I just mean like first derivative, second derivative, third derivatives and highest derivative Appearing is called the Order of the Equation. Okay, so this is just some introductory terminology differential equation. In the broadest terms, it's just an equation that relates a function to its derivatives. And then the highest derivative is called the Order of the Equation, and I want to make one other note that when I save differential equation here, I really need ordinary. So that will. Maybe that doesn't mean anything to you right now. But if you take a course in multi variable calculus, you'll learn that when you have multi variable functions, you have thio deal with what's called partial derivatives. Because I have to make a choice, actually, about what the independent variable is, and so that gives rise to the idea of partial differential equations. But here, when I say ordinary, I mean that I'm on Lee thinking about functions of one infinite variable. And when I say derivative, I mean the derivative with respect to that one independent variable, just like we've been doing for this whole course. Okay, and then the highest derivative with respect to that one and variable is what we're calling the order. Okay, So before we get ahead of ourselves, let's just look at some examples. Okay, so here's four differential equations, so notice the first one is the first order because it on Lee involves the first derivative of the function. Okay, The second equation is also first order notice that I'm really kind of very reputation. So this is kind of this differential like exploitation. But it's still a differential equation because it relates why, to the derivative of fly safety by D. X, its first order. Because the I d. X, that's just the first driven. Then this third one is a second order. Now, why is it second order? Because the highest derivative I see is the second trip. I also have the first derivative, and I also have kind of what you could think about this zero derivative because I have a function itself. Why? But this is the second one because the largest or the highest derivative is this. And then this one is the third order because the highest derivative is there a triple prime of X equals five x so pretty straightforward, just an equation that relates functions and this derivatives. And then we have this idea of order. Now, I claim that we've already encountered several differential equations. And up till now, the types of differential equations we encountered are what's called first order. Okay, so you know what that means? Separately? Equations. Okay, so we dealt with this like I said, Ah, lot in this course. Basically this was just finding anti rivers. So a separate ble first order equation can be written in this point g of y to some function of the dependent variable y or of the function times B y equals eight your backs DX and now notice. To start with, we may start with something like de y the X equals some function of X and y And so to say that this equation is separable means that I can write it in this form and this is not always possible. Okay, so not all equations are separate. In fact, let's see if you can find them here. So you know. So, first order, this one will be separable, so I can give you an example. So, for instance, if I have you, I d X equals, say sign of X times Why? Or something like that? You see, I can't just put all the things that involved why on one side and all the things been involved at some other because sign of X times why? It's sort of stuff. But you know, if this waas so this is not separable. Here's an example of a function that is separable if the I d X equals X time slots, this would be separable since I could just multiply by d X and divide by why and get one over Why? Why equals X times DX? Okay, so I can write It is a function of Bligh times do I and a function of X t X And the reason I like separable equations is because I really could just solve them by finding anti drip so I could just integrate both sides. And then what I get is g f y equals h of X plus c. We're obviously G A. Y is just an anti derivative G and h of X is an anti derivative h Okay, And then we just have our integration constant plus right, because he's anti derivatives, both will differ by some costs. I don't know what that is, so that's kind of the general solution of a separable first order differential equation. Okay. And we basically saw this idea in taking anti derivatives. Okay, so this is kind of the connection between what we didn't count coup and and sort of introductory differential equations. Okay. So, like I just said, we end up with this plus so solving differential equations need to Constance that are arbitrary. And so there's kind of a general rule that whatever the order of the equation is, that's how many constants you're going to end up happy. Okay, So for our first order equation, you're gonna get that constant and immigration inseparable that, plus seats. So, you know, we're really interested in finding unique solutions to differential equation. But the problem is, we get these integration constants that are arbitrary. So we actually get classes of solutions. Okay, so we get a bunch of solutions that differ just by these. You can think about them is constants of immigration. So to find unique solutions, we mean what's called initial there, sir, for instance, wives, zero crime of zero. My double problems here. Seven. And so when we were finding anti derivatives to get a unique anti derivative, we needed a point on the graph of the function. That's basically what we're doing here. So we just given points on the craft graphs of these options. And now physically, there's another way to think. You know y zero. Why private? Zero double come zero. So you can also think about these as and this sure positions of particles. So, of course, where I end up later depends on where I start. Depends on initial velocities. Is that okay? So these this initial data really can carry information about the system that the differential equation is described. But the connection back to what we did calculus is that when we found the anti derivatives, we've got constants of integration, and we needed some of these points to be able to figure out what those constants were. Okay, so that's more or less the idea. Okay, I know That's kind of yeah, that's probably a lot of information summarized very quickly. But the last thing I want to say is that this sort of problem of finding a unique solution to a differential equation using initial data is called initial value problem. So I the Okay. Okay, so that's maybe the first couple weeks of the differential equations class just slammed into a few slides. But that's really all that we need to introduce to be able to get into some really interesting examples. So let's actually do an example of a problem that requires solving an initial value problem. Actually, multiple initial value problems to really get information about what's going

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