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.