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Problem 4 Medium Difficulty

(a) For what values of $ k $ does the function $ y = \cos kt $ satisfy the differential equation $ 4y^{"} = - 25y? $

(b) For those values of $ k. $ verify that every member of the family functions $ y = A \sin kt + B \cos kt $ is also a solution.

Answer

(a) $k=\pm \frac{5}{2}$
(b) $y=A \sin k t+B \cos k t$ is a solution of $4 y^{\prime \prime}=-25 y$

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Video Transcript

someone find values of K that satisfies the differential equations here. So, you know, we need our regular Why equations which we're in a caf. But we also need to find our wardrobe formulation so right, it's like the direction function here. But first, we need to remember that the derivative of co sign is negative Sign and the derivative of Sign is positive co sign. They really use those. I didn't notice when we take our derivatives, So what? Crime peoples. So using chain room, we take the different movie outside, so co sign changes to negative side. They believe the inside the same times the derivative are inside, which is cake those of a part of this. Now it's upon our wide double prom equation. And remember that kay is a constant. So he's still using train. We're not product. So the take the derivative, the outside. So that's gonna be negative. K co sign, leave the inside the same and times that review of the inside chosen by this into that, we're going to use all three of these equations, and we're gonna put those into our differential equation here. So we are four times are wide open prom equation, which is negative case. We're co sign Katie and said it equal to negative 25 times our original. Why equation? It's ah, they're distributing. We're going to get this and we can add 25 something. Casey, the boat side. So we're gonna 25 co sign. Okay, T minus four. Case, we're concerned Katie equals zero, and we want to pull out. Look of sign. Katie. So we're left with 25 minus four. Case weird on the inside. And we never lived with these two factors. We're gonna That was inside. The presidency is equal to zero. Everyone, it's all okay. Okay. Equals five over two, but also equals native five over to because it's a square root. So we can't forget about that negative for these arcane values. And we're gonna need these four part beat the Barbie esus two. Bear by every member of the family functions. So what are your goals? A sign. Katie. Let's be co sign Katie and one of their part. It is also a solution. Er the y four times. Why? Didn't find equals negative 25 times Our original equation of what? So we're gonna use our values that we found in part A. Okay, so first we're gonna plug in. There is a K equals positive Bob or two. I want to close that until there is no more equations. And we still need to be able to use this equation here. So we need to go all the way up to a lot of fronts. So first, let's take the derivative. So a is a constant. So So using trained rule, take the derivative of the outside foreign goes to co sign, leave the inside the same times the derivative of the inside, which is five over, too, and do the same thing here. He's a constant so interactive horribly outside. Your co sign is negative. Sign Leave the inside the singing This is gonna be negative. Be so on times that originally inside, which is both ups. So we simplify this into there's here answerphone one double prime. We're going to the same thing. So we take the ribbon on the outside, which is negative. Sign. Leave the inside the same times that review on the inside. So hell of a lot of times. Negative. Bob lover too. We'll put parentheses here. Do something here. So subtracting. Take the roof of the outsider of them Signs co sign, Leave the inside the same times the drift of the inside. It's a renewable supply by father Over to There was simplified this since a negative 25 over four. Thanks. And so I think here. Negative 25 over four b goes on Loveline Tootie. So now we have all the equations that we need for to plug into our differential equations. So we're gonna use thes They were just gonna substitute them in. So, you know, for times are blue, which is our wider were firm. I resent that equal t o negative. 25 times our original law equation. I don't think this all fit on one line, so I understand. Yeah. Then I would distribute they were left, Would it? No. You're 25 a just distributing that for two for a look inside. And I'm going to shoot this 25. And so our right hand side and our live in zat is exactly was on our right hands on their four. We verified that solution for their value. Okay. However, we need to do the same thing. Four when k equals negative bar tabs so soon have the same one equation. So what I was gonna equal Hey, times sign of negative body, half sti plus b times co sign of negative five bean. See, they would still want to use the differential equation four times wider. We're fine. Is equal to negative. 25 times are y equation. So then do something here. Gonna take a lot of pride subversive on waterfront. So take the derivative the outside Leave the inside the same times the derivative of the inside, which is negative five over into do the same thing Here are the review of the outside is negative B spine. Leave the inside the same times in to remove the inside, which is negative. Five over too. So all that's just followed So simple by that Well, this here ever wide over Prime, who have our rejected over with the outside. Don't give us a positive father over to a sign. Leave the inside the same. But now it will supply buys that revealed the inside which is negative. Five over too. And then something here. So we have five Has the negativity outside the ghost co sign leave the inside the same time too derivative. The inside, which is negative. Five house. So we simplify. This will get positive. 2500 for a sorry negative. Nasty. Minus 25 before the co sign. Negative five. Over to tea. Look. And these the equations. We're gonna use a plug into our differential equation here. It was given so bored. Times are wide open formulation and for our right hand side will have negative 25 times our original. Why equation? So now we have to do to simplify by distributing you will have this here is equal to negative 25 distribute that into our right hand side. Okay, I made a mistake because we have ah, negative here on a positive over here, they should have the same signs. So let's see where I made a mistake. Uh, okay. Did you really with Let's look at our war prime find the coastline is a positive terms. That negative negative coast on the sign is negative times. I negative. No, Still make it positive. So something like that, right? So it's looking r wardell a prime. You have co sign a sign that would be a negative, which would make the whole thing positive. Times are negative inside. So this is actually a negative here, so they'll be negative here. Don't mean to be negative. 25 signing. So since our left inside is exactly it was far right hand side, we bear by this value. Okay, also.