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(1 point) We consider the Euler-Cauchy equation r"y" + 2'y" 37ry' + 72y = 0. The auxiliary equation of a general Euler-Cauchy equation ar&q...

Question

(1 point) We consider the Euler-Cauchy equation r"y" + 2'y" 37ry' + 72y = 0. The auxiliary equation of a general Euler-Cauchy equation ar"y" + br?y" + cy' + dy = 0 is given by ar(r 1)(r 2) + br(r 1) +c 4 d = 0,(1) For this problem find the auxiliary equation:(2) Find the roots of the auxiliary equation: separated list )(enter your results as a comma(3) Find a fundamental set of solutions 91, 92, 93 results as comma separated list )(enter your(4) Find

(1 point) We consider the Euler-Cauchy equation r"y" + 2'y" 37ry' + 72y = 0. The auxiliary equation of a general Euler-Cauchy equation ar"y" + br?y" + cy' + dy = 0 is given by ar(r 1)(r 2) + br(r 1) +c 4 d = 0, (1) For this problem find the auxiliary equation: (2) Find the roots of the auxiliary equation: separated list ) (enter your results as a comma (3) Find a fundamental set of solutions 91, 92, 93 results as comma separated list ) (enter your (4) Find the general solution with arbitrary constants A,B,C



Answers

Determine an integrating factor for the given differential equation, and hence find the general solution. $$\left(3 x y-2 y^{-1}\right) d x+x\left(x+y^{-2}\right) d y=0$$

Well. Day, ladies and gentlemen, today's problem or today's final problem, I guess, is to look at a couple problems. Higher order. Ah, higher order problems on. We're supposed to try and emulate the what happens in a double root case in the first sleep in the first, um, in 4.2. Um, if you remember, if you have a double route, um, then you ah too, um unique. Horrid to absolution, Czar. Each of our t plus t each of the rt. So that's what happens if you have a double root of our right. So we want to do the same type of thing, but in this case is a higher order. Um and so we're gonna try to emulate the same idea. And so, uh, let's look at the first case in the first case, of course. Turns out that the auxiliary equation is R squared, plus one squared S o. The roots of that are plus or minus. I repeated. So, um so that means that there's four routes, but I am negative IRA balls repeated, and soon our goal is to show that this this'll ce really looking thing here at the end is in fact, our general solution. And of course, to do that, I didn't Really I sort of short cut something here. But you should technically first show that these are literally of these functions are all in early independent, so that would be your first case. But I'm just gonna take it that they are and not bother showing it. But you you should technically show that they are, um, literally independent. Um, Okay. And And so the other step, of course, is once you know that those air all literally independent well, you have to know that each are a solution. You'll know that you'll notice that if each of these are solutions, so we have one, 23 Yeah, linearly, independent terms. If they reach a solution, then since this is 1/4 order ordinary, different equation, then this would be our general solution. So show that they're each solutions. Now, basically, all you really have to do is, um, toe actually calculate beach derivative. And then, um, where this guy's not Ah, it's got a sort of hiding here. Uh, but, uh, I'm sorry about that. Right. Well, that's folks, um, I wanted Okay. Why isn't it doing it for me. Um, not behaving properly. Okay. So essentially, what this stupid thing is supposed to say is that you plug each of these their radio. Okay? I didn't want to do it. Plug their requisite equations or the requisite functions into this, and you'll find that, in fact, they some to zero. So that they are a solution is to key. Okay. And essentially, you want to do the same thing over here with tea co signed t, and you go through and do the same thing. And again you plug. I did the derivatives here, but once you do it, then you plug it back into here and you'll see that. In fact, um, they are solution. So turns out that, yes, each of these for their linger the independent for solutions. So that's the general solution. Oh, in the final case, 37 be here again, and I sort of short, but the whole thing, and so it's a bit of work, but, um, the other books, huh? You can do it basically. And so the thing is, is, first off, um, we get the roots of the auxiliary equation. Negative one plus or minus square root three I and these air again doubled. So So these are double roots. And so that means that the general solution is again very ugly. Um, not a fun beast at all. And in this case again, what you have to do is first off, um, shows that these air literally independent functions, which is, you know, great. And then you have to show that they're everywhere. They're each of those air solution, so I'm not really gonna bother doing it, But that's what you'd have to do. Um, so sorry about that. I'm just Ah, uh, yeah, I just But that's what you have to do. So it's just a lot of work, so I'm not gonna do it here, but that's that's that's what you have to do. And, um, I'm not even clear in this case how you exactly go about so showing that they're literally independent, even though they are sorry, I don't see the answer right off. So Okay, so that's sort of the, um that's sort of the gist of how to do this. Thank you very much. I apologize for the shortness of video, but it was either cut it short or they would not do it completely. So the least I thought I'd try and do it. And if a brief explanation of how to do it, um, but thank you for your time. Um, but hopefully least gave you an idea how to solve it.

Hello? Today we're going to start robber number eight from the section tapped out of you here, we had to find the general solution of this different prosecution differential equation. Given if why number less bless their basic for wireless plus 26 by equals, Phil. So replacing wide, urbanizing. And they killed plus Larry in the square. But that he thinks the last 20 things inquires, Sil, you're given us minus one is one of the root food gangrene. And as deep Bressler and the remaining routes are the square pressed attendee Last 26 equals Phil So B equals minus one minus five minus. I minus Phi best I. So the solution gender solution can return us. Why off, x? Because you with this minus a plus off minus B I then the solution is dearest to X because x all be expressed Eagerness to eggs sighing being so we can write this learners President Carter. See you are you did a part of my an ethics. Plus I think so. Do you put the ball, uh, my left by next? Because it's plus stand ready. My enough. My ex fighting. They use the impersonation. Autistic for execution. There's enough Our question. Thank you

Here in this problem, we have toe find a general solution for the given. A differential equation on the given differential equation is why double prime Makes a Z equals two X divided by one minus X square through the power three. Divide by two. Now to find in general solution will be invigorating the differential equation with respect to X, So let's do it. Integration. Why Double prime mix dot de x is equal to integration. X Divided by one minus six square to the power three by two Dark DX No ive elevate this Andrew. I'll use substitution matured and I'll be substituting one minus x squared. Nasty. So we'll get minus two weeks dx us Did he? Or we can say that X DX comes out to be a C minus city Dubai to so that immigration becomes X DX is minus. You do I do and minus one. My dorm dating outside on this is wonder one minus x squarest Be so two to the power three bite toe Overhand right disintegration as minus one by two. I'm digging this in. The new monitors will get minus three by two Dark beauty. I'll be using this formula integrated and evaluate isn't regal. That is extra. Bauer endured. Eat door DX physical two extra power and bless one Divide way and press one So the answer will be equals two minus one by two Due to the power minus one by two. Divide by minus one by two. Blessed see received an arbitrary constant So minus one murder in my national radio. Cancel out and we're left with two to the power minus one by two. Bless. See on This is a goal to Why Primex? Now we can substitute the value of P. It will be called to one minus X square root of our minus one by to bless. See on this is my Primex again We have integrated to find them early or biopics. So let's do it again. So, integration. Why Premix Nordiques? As he calls to integration, we can write this Thomas one divide by under route one minus X squared. It's when the same thing Not underst Plessy Also there. So Placide RDX. I'll be using formula integration one minus under loot a squared minus X squared physicals to one by a side inwards X ray. You have heard this integral it will be goes to here if you compare both the functions here is one So we can write the result as signing worlds X by once only to write So only exit worse A plus hair. Bring it See X plus C one where someone isn't arbitrary constant on this will be called to y affects. So this is the solution for the given differential equation. I hope you about the problem. Thank you.

Hair. In this problem, we have to find the general solution off the given differential equation. Which is why pray Empty as equals 23 Plus it about minus duty. Sort of find a general solution. Integrating the differential equation. Board the sites. So the integrative Inspector p So by Prime D nor did he is equal to integration. Three plus it about minus duty, not duty. So he ever get via fee on the integration of three is three D plus. Irritable minus duty. Divide by minus two. Blessed see their season. Arbitrary. Constant. It will be called to three t minus it about minus duty. Divide by two. Bless. See? So this is the solution. Or you could say, a general solution for the given differential equation. I hope you get the problem. Thank you.


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