Calculus: Introduction to Differential Equations

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07/16/2011

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you have'nt covered the entire course on that

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Differential Equations
Separable Differential Equations
Introduction to Differential Equations Page [1 of 2]
OK. So, what is a differential equation? Well, a differential equation is exactly what it sounds like. It's an equation that has derivatives in it. So, we're used to looking at equations like x^2+3=1 or x^3-4=7 or whatever. But, now the question is, "What about an equation that within it, has a derivative? How do you solve that? What does it mean to solve that?" Well, that's what I want to talk about for a while. I just want to get a vague understanding of what a differential equation is. So, let's just sort of get a quick overview of this realm of differential equations. It's very important, by the way, probably one the most important aspects of calculus, because many things in life and in nature and in science actually conform to certain differential equations. Movement in time, space, orbits and so forth; these things are all governed by differential equations. So, being able to understand differential equations is really an important and valuable thing. Also, in the societal world, if you're trying to analyze large collections of data or if you're trying to consider things that involve the psychology or the economics of something, a lot of times these are governed by basic models that involve differential equations, as we'll see.
OK. So, what's a differential equation? It's an equation that has a derivative in it. No big deal. So, let's just write one down. . There's the derivative. That equals something. Let's say it's e^x or =e^x. Now, this differential equation - this is a differential equation. So, you can impress your friends that you've seen a differential equation right there. But, what exactly does this mean? Well, it means that I'm thinking of a function y. It's the f(x) and the only thing I know about it is something about its derivative. I know that its derivative is e^x. Well, actually, I can figure out what the function is. I can do it just by inspection, because this is such a simple differential equation or you can think about it some sense integrating both sides. If you integrate the derivative, you just get the function y. if you integrate this, you just get e^x. So, you have to look at it by inspection or otherwise to realize that a solution is y=e^x. You can check. Is this a solution? Well, does this satisfy this fact? Is the derivative of y, with respect to x, e^x or =e^x? Yes, because the derivative e^x is itself. So, in fact, this is the solution. Now, is that solution unique? Is it only one solution or are there others? Well, remember when you integrate, which is sort of the undoing of differentiating, we introduce a constant. So, in fact, technically, there's a +C here. So, this is what's called the "general solution" to a differential equation. It's sort of the solution, where we don't know what this constant is, but for any constant you give me, this will actually be a solution to this differential equation.
Now, you can actually plot what these things look like. It's sort of interesting to do that for different values of C. So, let's try to draw some axis's here. Now, suppose for example, that C were just 0. If C=0, then in fact, I just have y=e^x. We know how that looks. That goes through the point 1. We have a left horizontal asymptote here with lots and lots of growth. This is y=e^x. Now, what happens if C were actually selected to be 1? Well, then I just take this picture and I move it up one unit. The entire picture would move up one unit. Maybe I'll use a different color. You have to use your mind's eye to visual what this would look like. It's the exact same picture, but just shifted up one unit. The orange is y=e^x+1. What if the constant were to be 2? I can pick any number I want in here and that will still be a solution. So, I should, in some sense, look at the whole family of them, look at the whole thing. So, if I put in 2 that would look like this. If I let this be 2, it would look like the exact same curve as before, but just shifted up into the (x+2) and so on. What if I were to add -1? If I were to add -1, I would take the original curve and shift it down one unit. So, it would look like this. This is y=e^x-1. What you see is that I'm producing a family of curves. In some sense - well, not really. But, they're all kind of parallel in the sense that they all sort of go along their own individual curves. They never intersect. They never cross each other. They just go along like this. These are thought of as solution curves. So any of these curves - you can imagine a lot of them. These aren't all of them. In fact, even in between, if I let a C= . So, there are tons of these things. You should think of lots of them all over here. The totality of all of those things produce what are called, "solution curves."
Now, if tell you a particular point, like I said to you, "I'm thinking of this differential equation and I'll tell you what y equals for a particular value of x" then what that does is actually forces us to find a particular solution. For example, let me just illustrate this. Let's suppose I tell you one more piece of information. This is not something I figure out, but I'm told this. Suppose I'm told that when x=0, y=2. Well, if I know that point must be a point on my solution, then that will allow me to solve for C. In fact, let's just think about that. If I let y=2, that has to equal e^0+C. So, I see that 2=1+C. I can solve for C. Therefore, 1=C. So, for this particular solution, what I see is C=1. So, for the purple, I see y=e^x+1. That's the particular solution that not only solves this differential equation, but also satisfies this condition, sometimes called the "initial condition," because I'm letting x=0.
So, what's the solution to this? Well now the solution would be which color? It would be this one right here, the orange one. The orange is now the unique solution. If I don't specify a particular point, notice that at 0, the thing is at 1, too. So, it goes through that particular point right there. If I didn't specify a point, then it starts this whole family of solutions. These are called solution curves. So, anyway, there's sort of the nuts and bolts of differential equation, at least a very simple one. You try to solve it and you actually get a family of solutions. If you have a particular value that the solution must satisfy, that will particular anchor down that constant and we'll get one exact solution, rather than the family. But, sometimes we're just satisfied with finding the family.
Now, another thing that differential equations have is order. Now what's the order of a differential equation? Well, it's just the greatest number of times that you are going to differentiate. For example, here this is a differential equation of order of one. You can say, "I only see derivatives to the first power." Here is another example of a differential equation. The ^, all plus 2x , minus y^2=sin x. That generally is a differential equation, because it's an equation that has derivatives in it. That's all that's required for it to be a differential equation. I see an equation, the derivatives. So, it's a differential equation. But, the order of this differential equation is actually three, because the greatest number of derivatives I'm taking throughout this thing is this right here. This is an order of three. OK. So, that's what order basically is.
Here's another example of a differential equation. =e^-y sin x. This would be an order of one, because I just have a first derivative there. That's another example of a differential equation.
OK. Now, how do you actually solve differential equations? How would you solve this differential equation? Well, the answer probably is that it's very, very difficult. In fact, differential equations, not surprisingly, are sort of integrals, because if you want to solve a differential equation, like this basic one, it sort of involves taking an integral. We already know that taking integrals of exotic functions, that's serious business. So, it's not surprising finding the solutions to differential equations is very serious business. In fact, there are courses on the subject. People just take a whole semester, a whole year of study or their whole lives, just to the subject of solving differential equations. This is serious stuff. So, something like this might be a little bit difficult to tackle right now.
Some times we actually can look at. For example, let's look at this. This one looks a little bit more threatening than the first one we looked at, which was this. That one we could do just by inspection. Now, I've got some y's in here. Let me just try to make an inspirational observation about this. Then we'll see how to really proceed when we see such equations. So, if I want to look at this, let's just perform some fantasy math here. Let's see if we can just have some fun with this. You see, one thing I can do, if I really wanted to be silly, is to break this up. Now, technically, this is the derivative of y with respect to x or . It's a symbol that means the derivative. Let's think of it as a change in y over the change in x or in fancy words, the differential with respect to y over the differential with respect to x. Don't worry about that. All I'm saying is think of it now as a fraction. I think of it now as a fraction. I fantasize about multiplying both sides by the dx, but notice the dx will cancel here and I get it up here. So, what I would see is dy=e^-y sin x dx. So, I would see that. If I see that, notice that if I divide both sides by e^-y or multiply both sides by e^y, then it would cancel out here and I would get that on this side. E^y dy=sin x dx. Now, I happen to like this form of this particular differential equation, because notice that now all the y's on one side segregated from all the x's that are on the other. I like this, because now that I split them off into an x-group and an y-group, maybe I can just literally integrate both sides and actually see if I can solve this differential equation. Differential equations of this sort are known as separable differential equations.
Anyway, I'll let you think about if you can solve this or not. I'll see you at the next lecture.