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Show that $u(x, t)=f(x-c t)+g(x+c t)$ is a solution to the wave equa$\operatorname{tin} c^{2} u_{x x}-u_{t t}=0,$ where fand $g$ are smooth functions.

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

Campbell University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

04:14

03:42

Show that the functions ar…

12:25

Let F and G be arbitrary d…

07:16

(a) Show that the function…

03:07

04:31

13:06

Yeah. So in order for us to show that this is a solution to the wave equation, we'll first need to find the partials. Then we could come up here and just plug it into here. Um, so let's find those partials first. So if we do the partial but respect to X so we would take the derivative of this function F first with prospective exit would be f sub, x, uh, x minus C t. And then we do del by Dell X of X minus C T and then plus the partial of G with respect to X so g x x plus C t. And then the partial of this with respect X. So now remember, we're assuming that T is a constant and see is also going to be a constant um so when we take the drill of X minus ET with prospective X, that would just be one and the same thing over here for X plus c t. One. So that would give us f sub x x minus C T plus the partial with prospective G or the partial book perspective, X f g Extra City. Now we can go ahead and take the next partial of this. So we would need to do chain rule again for both. So now will be the second partial and then X minus C t. And then we would give our multiply by the derivative on inside do the chain rule, which is just gonna be one. But then over here, same thing. So second personal X plus C t multiplied by a derivative on inside which once again we already found was one. Next let's go out and get rid of this. Now let's go ahead and do the same thing. But for t so you X is going to hope not you x u t is going to be so partial of effort, respect ity of X, my C t and then chain rule Dell by dlt of x y c t plus g so t X plus C t and then del by Dell T of explosive et Now we're taking the derivative with respect to t here. So what we assume acts as a constant. So this here would just be negative. See, then over here this is going to be seek, So let's go ahead and write that out so negative. See partial of effort. Prospective t next, minus C t plus C times the partial g with prospective tea exports ET and then we go ahead and take the partial of this again for perspective. So be the same thing. But we just get the second partial here and then derivative of the inside same thing. Give us negative C waas. See double partial T X plus C t and then chain rule. You have a multiple. See out here. So let's go ahead and rewrite this. Actually, those cancel out with each other and then notice how we have a C squared. So I'm just gonna factor that out for it'll be apparent. Why won't do since I get so b c squared times of the second partial of prospective t of X minus C t loss G second partial derivative with respect t of X plus C t. All right, so now we want to look at this equation here, so c squared. You xx minus u t t which, if we come back up here and look at what, um, are partial with respect to X is, um it's just both of those being added together. Yeah, So this will give us so c squared times f of x x x minus C T plus G of my duty G second pressure with respect to X of exports city And then this would be minus c squared. I should think I forgot parentheses right here of f tt X minus C t and then plus three of TT X plus C t knows we have the seesk words for both of those who would go on a factor that out. So I'll write this little bit further off on the right side of some more room, and then I would just copy all that down again. And so now we want this to be equal to zero. And remember, this is going toe hold as long as it holds for some value of seats. Um, I don't know if they actually say this anywhere, but normally when we're looking at this wave equation, this value of C is going to be a, um, non negative value. So one value that will make this true is if we use zero, um, for seat, and so we could maybe have some, like weird things were like these partials here. Cancel out. And then these partial here. Cancel out, Um, but without given without us being given more information is kind of hard toe, actually. Say, if anything can be done like that, Um but we confer. Sure, at least say Well, so if C is equal to zero, then we have zero times, You know, all of this, which would be zero, and then that would satisfy the wave equation. Um, and so if you're kind of on sure about this, we could also just go back to the start and just say, Okay, well, let's let's see B zero and you'll see how we'll end up with this. Um, so actually, let's write this out first. So C squared, you xx minus u t t. So again, if c zero then this is just going to leave us with negative U T T. And we want so want this to be zero. So, in other words, the second partial retrospective t is just zero. So let's actually look to see what the second partial of this will be. So we end up with. So if we have you, x of t is equal to f of X minus city plus G of X plus C t. And again, we're doing C is equal to zero. That is just going to give us ffx plus g of X. And so if we even look at the first partial of this with respect to T, that would be zero second partial. That would be zero was well, so you can see how that satisfies what we want up top there. Um, yeah. So all this isn't really necessary down here, just kind of showing what we had. It actually does work, even if we start from the beginning with this value of C, um, because they don't really put a restriction on C as long as it is non negative. Yeah. So this is how I would kind of go about it. So So you just kind of plug them into the wave equation, and then you make this observation. Oh, we could factor off the sea. And then it's either this thing is zero or C squared is zero. And again, we fell it some extra information. We can't necessarily say anything, Maura, about this expression. Um, so that's why we can just choose arbitrarily see, to be zero, and then just have it be equal to zero that way

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