5 answers

(Li points)_Ware_Solution (AlLor Nothing) Demonstrate by direct calculation that the function u(x,t) = f(x -at) satisfies one of the

Question:

(Li points)_Ware_Solution (AlLor Nothing) Demonstrate by direct calculation that the function u(x,t) = f(x -at) satisfies one of the three equations given below Satisfies" means that the substitution of ulx,t) = f(x- at) makes both sides of the equation to be equal ar2 at?


Answers

Consider $f(x, y)=\ln \left(x^{2}+y^{2}\right) .$ Show that $f$ is a solution to the partial differential equation $\frac{\partial^{2} f}{\partial x^{2}}+\frac{\partial^{2} f}{\partial y^{2}}=0$

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Let us have those functions it affects why is execute minus five X. Y square. We want to check if this function is a solution to the to this partial differential equation. Okay, so X times to cross partial derivatives minus uh minus, particularly with respect to why it's a good 20 So we're just gonna put this function in this form and see if is going to give you zero after the this operation. So you're gonna take it at first partial derivative with respect to X.

So this is gonna be three X square. But this time you should be able to do partial derivatives, right? Otherwise you can watch the previous tutorials to see how we do it. So this is just gonna be straightforward uh five Y squared. And then I'm gonna do F X. Y.

This is fx. So just to get partial derivative with respect to Y. And you're just gonna get negative 10. Why? Because this is a constant. Uh What is F.

Y. Well that is negative 10, right? X. Y. As well? This is supposed to be why? Okay, so now we're gonna put it in this form, So x. Now this one, is that okay? So that is going to be negative 10 Y.

So in fact gonna put it this way it's gonna be negative 10 X. Y. Right? Because this is X. And this one is this one so it gives you that then uh minus. Now this is F.

Y, F. Y. Is this guy? So I'm just gonna put it in place of it. Yeah. Now what does that give you? Well, this thing is negative 10 X Y minus negative, negative is positive is gonna plus, right, 10 X.

Y. So negative 10 X Y plus 10 X. Y. Is going to give you zero. So you can see that uh this guy is giving you zero.

So it means that it satisfies this equation, right? Because this one is zero and this one is zero. So the portrait differential equation is satisfied.

So in the human question there is a differential equation degree by the access it was to wait and why equals to affect the solution to the equation. We have to check whether Y equals to do affects the solution or not. Like So from here, from these two we can say that um, FDA checks S equals two effects, right? No, let's take Y equals to do effects. So divide by dx will become to wear flash X. They'd Now when we create the values of the debate by DX We'll see that to FDA checks is equals two.

Mhm. To fx. Yeah. Right. Mhm.

Yeah. It means since Alleges as equals to ologists. Yeah. Therefore he has since after taxes equals two Effects. Therefore Alleges equals two largest like therefore bike was to do effects is also a solution okay to the differential allocation like.

In the given question we have a differential equation. Do you have by the X. S equals two X. And it is given that Y equals two effects is a solution to the differential location. So now we have to determine whether why it was 22 FX is a solution or not.

Right. So from the given a question we can see that F dash X is equal to X. Right? Let's now let's take the function Y equals to do effects. When we differentiate both sides, we get White ash is equals to do after sex. We can replace aftershocks by X.

Y equals Y. Dashes equals to the works. They Now when we create the White Ash. Mhm. From the differential equation, since divide by the XS equals two x.

So here it will become X. And the right hand side will become two X. Since X is not equal to do X. It means alleges is not equal to a biologist. Right.

Therefore Why equals to two effects is not a solution. Yeah. Mhm. Yeah. Right.

Yeah..

We want to show that these three are solutions of our differential equation, so we could just take the derivative of each of those three functions, plug a man and go from there. But one thing that may help us cut down on a little bit of time is if we solve for C first and then use that to answer a and be released to show that it holds current. So let's start with C. So let's see. What? Why prime ISS or C? So we would write so negative deep I d.

X of X plus seat raise the negative first so we would use power rule to take this derivative move the to outfront. Subtract one off of it. The negatives cancel out so that it's just going to be X plus C rays to the negative second, which we can reciprocate as one over X plus e to square. All right, so that's what Why pry Ms. Now let's see.

What why squared is so why squared is Nega Do one x will see squared, which is just going to be so the negative goes away since we're squaring it and it speaks one over x plus e square. So why squared is equal to y fronts? Coursey, Right? So see checks out now for parts A and be actually. So for part, A A is the case. So it's heart. See when C is equal to zero so that one will check out as well.

And B is part see when C is equal to three. So by just showing it holds for see first, it's Trivoli follows that A and B work as well..

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