Calculus: Solving Separable Differential Equations

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Differential Equations
Separable Differential Equations
Solving Separable Differential Equations Page [1 of 2]
How do you solve differential equations? The answer is, in general, it's very difficult. But there are some special differential equations for which there are techniques, which allow us to actually break the differential equation wide open and find out what the function is, for which satisfies the particular differential equation. So, here's an example: =e^-y sin x. Now, how would you look at this? Well, I notice that this equation can be separated in the following way. If I imagine multiplying through by the dx, by the differential with respect to x, I see dy =e^-y sin x dx. Now, if I multiply through by e^y, then it cancels here and it comes up here. So, I see e^y dy=sin x dx. Notice what I've got here. I've got some function that just has y's in it, times dy, equaling some function that just has x's in it, times dx. Whenever you can separate the variables like that, what you have is a separable differential equation. So, here's an example of separable differential equation. I can separate the x's and the y's. When you can do that, it's actually really great. In fact, differential equations of this form aren't usually that hard to solve, because all I have to do now is literally integrate both sides.
So, how do you now solve this particular separable differential equation? Well, what I do is I literally integrate. I'm in good shape, because I've got the dy here and I've got a dx here. If you're going to integrate, you've got to have a differential at the end. Don't forget that. So, you add a dy or dx or whatever. Now, the integral here, that's just e^y. The integral here is -cos. Now, technically there's a constant +C here and a constant +C here. I want you to think about. They might be different constants, too. But, if I take this constant and bring it to this side, then I have one just huge constant. So, what I tend to do is to just write this as a +C. Understand that C is actually both a constant here and a constant for this one, but I brought this constant over. So, I accumulated all the constants. Well, so there's an answer to this particular differential equation.
Now, in fact, you could actually solve if you wanted to for a y. If you wanted to get y alone, all you would do is take the natural log of both sides. You take the natural log of both sides and you would see that y=ln(-cos x + C). So, that's a solution. That's a general solution. You have to know what the constant is. So, in fact, if you vary the constant - in fact, this function will be different and be changing. You can actually graph this and see when you change constants, what happens to this particular picture. What you see, by the way, is this. So, this is the solution curves for this particular differential equation. That's the modified one. Here's the original one. So, when you solve that, what you see is actually this. As you vary C, each of these lines that you see here, each of these curves, represents a different value for C. When you fix C to be some particular value, then you get one particular curve. As you let the C's vary, you get this family of solutions. You can see what this thing is doing for different values of C. If you actually want to solidify a particular C - for example, suppose we're given the following fact. When x=, y=2. If we put that in, then we could actually figure out what C would have to be, which particular curve we're relying on. How do we do that? Well, 2, that would be y. Y=2. 2=ln(-cos + C). What's cos of ? Well, that's 0. I see that 2=ln (C ). What does C equal? Well, if you exponentiate both sides, raising the powers, untangle this, a log is an exponent. What I see is that C=e^2. So, the constant in this particular example with this thing fixed would e^2. I can come back and plug that in. What I see is a particular solution, given this particular case, would be ln(-cos x + e^2). There's a particular solution, given this particular condition. So, if x=, y=2. So, , that's about - this is 3.14. So, this is a little bit bigger than 1. So, when x is a little bit bigger than 1, y is 2. So, when is x is a little bit bigger than 1, y=2. That's way up here. We don't have that particular level curve drawing, but you can see what's happening. It would be something that would sort of come down and dip up and go through that point and go right along there. So, it sort of captures this thing. The graph of this is basically that. That's what we're seeing. OK. So, that's pretty neat.
Let's try another one together. Separable differential equations. How about this: =? Now how could we resolve this? Well, the first thing I would actually have to ask myself is, "Can I separate these variables?" Can I write it as something with just y's times the y equals something with just x's times the x? If I can do that, I might be able to crack this differential equation wide open. Let's try. If I multiply through by the differential dx, it drops out here and I get a dx there. So, that's a good start. What I would see is dy=dx. Now, see if I multiply everything through by the (5y+1), it will cancel here and appear there. So, what I would now see - I can modify this to be (5y+1)dy=3xdx. Notice that I really have now stuff with y's times dy equals stuff with x's times the x. This is now verified to be a separable differential equation. I separated the variables like that. So now, actually solving this is not going too bad. All I'm going to do is integrate this side with respect to y, integrate this side with respect to x and see what I get. So, if I integrate 5y, I see , integrate 1 and I just get y. Technically plus a constant, but I'll save them all up for the other side. So, this equals - what's the integral of this? The integral of this is going to be x^2+C. The constant of integration for this and then minus the constant integration for that when I bring it over. It's just one big old constant we don't know. That's the answer. That's the general solution for this particular differential equation.
You might say, "Gee, should we solve this for y?" Well, you might think that, but the answer is you actually can't solve this for y, because you get something that's not really a function, like hyperboles and so forth or ellipses. They're not functions. So, in fact, we can't - we have to be satisfied with this answer and realize that we can't actually solve this for y. In fact, if you look at the level curves, if you change C, if you vary C and see what happens, you actually get these pictures. So, you get this sort of hyperbole-looking things. These are some of them, of course. There are a lot of them. They fill up all of space, in fact. So, this is just a few pictures, but it allows you to get a sense of what's going on. What this means is, if you pick a particular constant, then you're going to get a particular curve here. In this case, it's going to be a pair of them, because these are hyperboles. In fact, you can see a visual representation for all the different family. They are not really parallel, but they sort of go together in harmony like this for the different solutions of this differential equation. This general solution provides all the solutions. If someone gave us a particular value, like when x=0, y=1, I could plug in 0 and 1 and figure out what the constant is. So, you can find a particular solution if you know a particular point. If you don't, you have to be satisfied with a general solution and realize that as you change the constant, in fact, it gives you different pictures like this. Neat. Separable differential equations. Neat stuff. See you in the next lecture.