Exact equations and integrating factors
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Integrating factors 1
A lot of what you'll learn in differential equations is really just different bags of tricks. And in this video I'll show you one of those tricks. And it's useful beyond this. Because it's always good when, if maybe one day, you become a mathematician or a physicist, and you have an unsolved problem. Some of these tricks that solved simpler problems back in your education might be a useful trick that solves some unsolved problems. So it's good to see it. And if you're taking differential equations, it might be on an exam. So it's good to learn. So we'll learn about integrating factors. So let's say, we have an equation that has this form. Let's say this is my differential equation. 3xy-- I'm trying to write it neatly as possible-- plus y squared plus x squared plus xy times y prime is equal to 0. So, especially since we've covered this in recent videos, whenever you see an equation of this form where you have some function of xy, then you have another function of x and y times y prime equals 0, you said, oh, this looks like this could be an exact differential equation. And how do we test that? Well, we can take the partial derivative of this with respect to y, and we could call this function of x and y, M. So the partial of that, with respect to y, so M partial with respect to y, would be 3x plus 2y. And if this function right here, that expression right there, that's our function N, which is a function of x and y. We take the partial with respect to x, and we get that is equal to 2x plus y. And in order for this to have been an exact differential equation, the partial of this with respect to y would have to equal the partial of this with respect to x. But we see here, just by looking at these two, they don't equal each other. They're not equal. So, at least superficially, the way we looked at it just now, this is not an exact differential equation. But what if there were some factor, or I guess some function that we could multiply both sides of this equation by, that would make it an exact differential equation? So let's call that mu. So what I want to do is I want to multiply both sides of this equation by some function mu, and then see if I can solve for that function mu that would make it exact. So let's try to do that. So let's multiply both sides by mu. And just as a simplification, mu could be a function of x and y. It could be a function of x. It could be a function of just x. It could be function of just y. I'll assume it's just a function of x. You could assume it's just a function of y and try to solve it. Or you could just assume it's a function of x and y. If you assume it's a function of x and y, it becomes a lot harder to solve for. But that doesn't mean that there isn't one. So let's say that mu is a function of x. And I want to multiply it by both of these equations. So I get mu of x times 3xy plus y squared plus mu of x times x squared plus xy times y prime. And then, what's 0 times any function? Well, it's just going to be 0, right? 0 times mu of x is just going to be 0. But I did multiply the right hand side times mu of x. And remember what we're doing. This mu of x is-- when we multiply it, the goal is, after multiplying both sides of the equation by it, we should have an exact equation. So now, if we consider this whole thing our new M, the partial derivative of this with respect to y should be equal to the partial derivative of this with respect to x. So what's the partial derivative of this with respect to y? Well, if we're taking the partial with respect to y here, mu of x, which is only a function of x, it's not a function of y, it's just a constant term, right? We take a partial with respect to y. x is just a constant, or a function of x can be viewed just as a constant. So the partial of this with respect to y is going to be equal to mu of x, you could just say, times 3x plus 2y. That's the partial of this with respect to y. And what's the partial of this with respect to x? Well, here, we'll use the product rule. So we'll take the derivative of the first expression with respect to x. mu of x is no longer a constant anymore, since we're taking the partial with respect to x. So the derivative of mu of x with respect to x. Well, that's just mu prime of x, mu prime, not U. mu prime of x. mu is the Greek letter. It's for the muh sound, but it looks a lot like a U. So mu prime of x times a second expression, x squared plus xy, plus just the first expression. This is just the product rule, mu of x. Times the derivative of the second expression with respect to x. So times-- ran out of space on that line-- 2x plus y. And now for this new equation, where I multiplied both sides by mu. In order for this to be exact, these two things have to be equal to each other. So let's just remember the big picture. We're kind of saying, this is going to be exact. And now, we're going to try to solve for mu. So let's see if we can do that. So let's see, on this side, we have mu of x times 3x plus 2y. And let's subtract this expression from both sides. So it's minus mu of x times 2x plus y. You'll see a lot of these differential equation problems that get kind of hairy. They're really just a lot of algebra. And that equals-- what do we have left? I'll write it in yellow. That equals-- I'm going to run out of space. I'm going to do it a little bit lower. That equals, just this term right here. That equals mu prime of x times x squared plus xy. And let's see, if we factor out a mu of x here, we get mu of x times 3x plus 2y minus 2x minus y is equal to mu prime of x, the derivative of mu with respect to x, times x squared plus xy. Now, we can simplify this. So we get mu of x times-- what is this-- 3x minus 2x is x. 2y minus y, so x plus y, is equal to-- and I'm just going to simplify this side a little bit-- is equal to mu prime of x. Let's factor out an x here. And the reason why I'm doing that is because it seems like if I factor out an x here, I'll get an x plus y. So this is mu prime of x times x times x plus y. x times x plus y is x squared plus xy. So that's why I did it, and I have this x plus y on both sides equation, which I will now divide both sides by. So if you divide both sides by x plus y, we could maybe assume that it's not 0. That simplifies things pretty dramatically. We get mu of x is equal to mu prime of x times x. And now, just the way my brain works, I like to rewrite this expression just in our operator form, where instead of writing it mu prime of x, we could write that as d mu dx. So let's do that. So we could write mu of x is equal to d, the derivative of mu with respect to x, times x. And this is actually a separable differential equation in and of itself. It's kind of a sub-differential equation to solve our broader one. We're just trying to figure out the integrating factor right here. So let's divide both sides by x. So we get mu over x, this is just a separable equation now, is equal to d mu dx. And then, let's divide both sides by mu of x, and we get 1 over x is equal to 1 over mu. That's mu of x, I'll just write 1 over mu right now, for simplicity, times d mu dx. I'm actually going to go horizontal right here. Multiply both sides by dx, you get 1 over x dx is equal to 1 over mu of x d mu. Now, you could integrate both sides of this, and you'll get the natural log of the absolute value of x is equal to the natural log of the absolute value of mu, et cetera, et cetera. But it should be pretty clear from this that x is equal to mu, or mu is equal to x, right? They're identical. If you look at both sides of this equation there, you can just change x for mu, and it becomes the other side. So, this is obviously telling us that mu of x is equal to x. Or mu is equal to x. So we have our integrating factor. And if you want, you can take the antiderivative of both sides with the natural logs, and all of that. And you'll get the same answer. But this is just, by looking at it, by inspection, you know that mu is equal to x. Because both sides of this equation are completely the same. Anyway, we now have our integrating factor. But I am running out of time. So in the next video, we're now going to use this integrating factor. Multiply it times our original differential equation. Make it exact. And then solve it as an exact equation. I'll see you in the next video.