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(1 point) Use the Laplace transform to solve the following initial value problem:y" 10y' + 41y = 0y(0) = 0, y (0) = Using Y for the Laplace transform of y...

Question

(1 point) Use the Laplace transform to solve the following initial value problem:y" 10y' + 41y = 0y(0) = 0, y (0) = Using Y for the Laplace transform of y(t), i.e; L{y(t)}, find the equation you get by taking the Laplace transform of the differential equationb. Now solve for Y(s_By completing the square in the denominator and inverting the transform; find y(t)

(1 point) Use the Laplace transform to solve the following initial value problem: y" 10y' + 41y = 0 y(0) = 0, y (0) = Using Y for the Laplace transform of y(t), i.e; L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation b. Now solve for Y(s_ By completing the square in the denominator and inverting the transform; find y(t)



Answers

Use the Laplace transform to solve the given initial value problem. $$ y^{\prime \prime}-2 y^{\prime}+2 y=0 ; \quad y(0)=0, \quad y^{\prime}(0)=1 $$

Uh this problem is y double prime minus two. Y problem minus two. Y equals zero. And we know that why is vehicles to and why promise zero equals zero. So if we start with the hospital sort of S square and possibly some of why my answering our best times Y zero at 2 s. Yeah my ass. And we have white primer 070. So it's going to not write the dancer and then move on to the next term to S plus. We'll start with them. Yeah. Mhm. The flash trance front of why Uh minus and I have to be distributed to the Y0 surrounding this of four. Last two options for why Equals zero. I'm gonna separate other terms why S squared minus two as lies to you go to and remove all the other terms the other sides to us before um Now I'm gonna isolate the loss transform so loss of why go to to as plus for all over. Um S Where Wise to S -2. So now we're going to do um I was gonna try to get this into a foreigner recognize so we can I can get this town too mm two plus for over. And then we can see from the s squared minus two with the pace of this has to be s last one squared since we have this negative to hear that we know that it has to be minus three. Um Okay so uh we know that we had to get a smartphone in the numerator So when you transform to s before and uh or S -1 into a space for us. We're not are essential pilots who is going to put you out here and distribute this, we would get to s minus two, so we still have plus fours, we're gonna add six. So we're gonna have is um well, I'm sorry. Right, tell you what, let's walk up. Yeah. So what we're going to have is uh two S plus one. I'm sorry. It's my last one. Yeah, it's plus six. That's going to give us that to S plus four our new mayor needs. And then so we're gonna uh all over S -1 Squared -3. It's not a separate is out. What what you have to outside of s minus one. This mask one squared one is three uh plus six. I'm sorry. Of six over. I'm gonna tell you that it's different. Six it Over S -1 square. The finest three. So that's where we're at now. Almost done. Not quite explain to simplify it a little bit further. So now we're going to have we know that we recognize this form as the form for a co sign the philosophy of coastline, but we know that uh just needs to be a square. All right, so let's change this two, two s minus one As much as one squared minus have square three square. So now we're gonna do the same on this side, but I'm going to factor the six or take it out, I'm going to take out six over square three because we know that we're going to need a script or three on top to make this work. Um Now I'm going to have to do uh S minus one squared Myers 33 squared as our denominator. But since this is negative on the bottom of the middle class, like you were going to make this negative top, so that's going to be uh that's what that's our most of the five version. It's very kind of y is equal to english applause transform of uh to hear S -1 Over. That's minus one squared my ass squared three squared uh minus six overseas, buried three As the plaster. This form of negatives Curtis three over that's minus one squared minus square 23 squares. Now you can see that why is just going to be equal to um do is uh this is the forms for the exponential function, multiplied by co signer assigned function. So the first one is going to be E to the T. I was co sign I believe negative Squared of three T -6 sq 2. 3 Times E to T Sign of Native Square three T. Is the answer to this problem

Because our problem here is why to prime minus two. I. Prime plus two, Y equals E. To the negative teeth. And what we're given is our wives here. zero. My problems vehicles one. So I started taking cross transfer of these terms of s. Where the box transform of why My S. Y0- Why? Promise 0? Oh no, no it's fine. Uh minus two. Best time to distribute this negative to arrive bones. Um Plus two. Why is you plus two? Fast transfer Y equals little flaws transform E. To negativity. Alright. So not only have that uh Y zero is zero so it's not go zero and that's going to go zero and I'll just enduring color Y zero Y. Prime zeros once you know this is going to go to one. All right. So so fast down, that's where loss transform of Y Uh -1 -2. S. Laplace Transform Why? Um Plus two classrooms from why equals as we know the fast transform of a exponential function is going to be won over S minus the coefficient on the T. So this is going to you uh S minus negative Gs plus one. Yes. Yeah. All right. I'm gonna uh further separate styles are gonna have plus that some of y. Outside of s square the mayas to S plus two and move the one to the other side sort of one over S plus one plus one. And so if we have given the one the same thing on there, we're gonna end up with one over R. One plus S plus one over X plus one. So that's just equal to Um s plus two Over. Ask this one. So now to get the transfer alone on the left side or I have a class transfer of Y. It's equal to as close to All over. s. plus one times S squared minus two X. Plus two. Oh and so from here we're gonna try to do a partial fraction decomposition. Which is going to be a over s. Plus one. Close B. S. You got messy. Um Must be at plus C. All over S squared minus two. S. Close to. Yeah so we're going to multiply A. By every term on the right hand and on air and ps received by S. 1st 1. And what that gives us is going to be a S squared minus to a. S. Plus two. A. and then the product of those B. S. Plus C. And S plus one is going to be P. S. Squared plus C. S. Plus B. S plus C. All that Is equal to our numerator which is s. Plus two. And I'm going to separate out into like terms we'll have a S squared plus B. S. Squared. You'll 20. I'll have native to A. S. Plus B. S. Plus CS. Well to pass around two A Plus the equal to two trying to divide out divide divide divide by S. And so we're left with A. Is equal to negative B. Negative to A plus B plus C equals one. And to a plus c equals two. Um She's double checking my work real quick to sleep too. All right awesome. Okay so now we're going to uh substitute and solve for these terms. So I'm looking I'm looking at this right this left hand term in the middle or left hand equation in the middle. So I'm going to swap um A. With negative beast. That's gonna become just to be because a single negative B. So if I have a a negative times negative two we have A. To B plus B plus C. Equals one. So three B Plus c. single to one um facebook. I'm actually gonna go the other rap. Now that's five out. So that that which I tried. That didn't work. So we're gonna go the other. I'm gonna swap in since I have an A. And C. Over here. I'm gonna try to just isolate this into an nsc. So I'm going to swap in uh negative A. For that beast. We have negative to a minus A. Plus C. Equals one. You mean -3 a. Plus C. equals one. The um And so I'm gonna bring that over here and I'm a lot of -3 a. Plus the equals one. And so I'm just going to subtract this second equation from a lot of times and they go to a Plus three. That was five a. Sees canceled Outside plus zero C. and two last 1 is one. Okay Now equals 1/5. All right great. So now we're gonna come back over here we have -3/5 plus C equals +12 C equals 1-plus 3 or five. So see 0 to 8 fists. So and if A. Is equal to negative bees and B equals negative once this. Alright. Scenario improvement please guys back into our uh first reaction the conversation at the top. This is now going to equal. Yeah. Mhm. 1/5 over. That's plus one. Close. Mm hmm. Negative on fifth. Yes. Was 8/5 all over. S squared minus two X plus two. Yeah. Mhm. Alright. So now these are informed that we can kind of recognize and solve for. So we're going to have that This is equal to 1 5th or one over. S plus one. And then I'm gonna look at this other side. I'm going to have plus and I'll have everything over S squared. I'm sorry. No. S my s one squared plus one. So does s square miles to s you know that that's going to be squared minus one which gives us S squared minus two S plus. Once we need to add another one to get catch that too. So in our numerator we now know we want to have an S -1 here And you know this is multiplied by a negative 1/5. So that would give us negative 1/5 ass plus 1/5. When you get to 8/5 It's going to add seven more 5 Scenario several itself further enough. 1 5th one over s plus one minus 1/5 S -1 over S minus one squared plus one plus 7/5 over one over S -1 Squared Plus one. So now we're going to solve for this, you know that why is equal to 1/5? I was a universal applause transform of one over S plus one minus 1/5. How's the university lost transformed of S -1 all over S minus one squared plus one plus 7/5. It has a little applause transform of one all over asked last one squared plus one. So Why is equal to 1/5? Eat a native T minus 1/5 E T. Co sign of t plus 7/5 E t T sign of teeth. There's your answer.

So our problem here is why to the four prime last y equals zero. And other than that, why is vehicles one problem? 00. Why that promise vehicles one and watch from 00. So they would start with this we're gonna start off taking plus transforms of each term. Are you? As to the 4th? That was the glass transform before I good uh minus s cube. Watch the zero my s squared Why prime of zero my S. S Why double from zero minus y triple prime zero minus applause transform of why equals zero. Sorry to cancel out terms. So we see that why primary zero triple prime zero are both equal to zero. Can cross off all these terms assuming zero And this is going to be zero and we see that these two terms of both people won. So we know that this term it's just going to be equal to s in this term is just going to be equal to s cube. It's not going to rewrite down here simpler to the 4th applause transform of Why Maya's s cute minus s minus laplace transform of why All equal to zero. So now it's much simpler when I separated out now on the left hand side we're gonna lost transform of why? Outside of S to the 4th um minus one. On the right hand side we have s to third plus ass. Sorry uh divide to um get the transform alone. Yeah, as to the third plus S All over us to the 4th. My ass one. All right, so now we have on top um we're going to try to simplify this out so we can do something with it. And so we're gonna separate our denominator first. So we said this is uh one is the other one square. So you see as a perfect square uh It's a product of S Plus one X -1 and fear that's called but this is only S Cubes Plus three of Top. I'm sorry. Plus ass you can go back. So if I was one further it's a little bit frozen right now. one Good. Let me do this thing. All right, we're back. Alright, so we're gonna tractor and that's on top. We're going to S. S squared plus one mm on the bottom where everything else is is the product of S squared plus one and essex where minus one. Now I have this on both top and bottom. She crosses out. Now all of a sudden we have just S over. That's Weird -1. Yeah. Which is a much easier. Um from deal with. So we're gonna give the denial of the same treatment again separated out. So you can see this is a product of S. Plus one & S -1. And as you might guess, we're gonna do a partial fraction decomposition. So take this down here have a over as plus one plus B. Or S -1. So now we're going to take it further and go A. S. My S. A. Plus B. S. Plus B equal to S. Sorry I separate out into like terms yes plus B. S equals S. I'm sorry yes never have negative A plus B equals zero. So from this we get that A. Is equal to be yeah and A plus B To go to one. So this is a pretty easy solved but so we get that A. And B Are both equal to 1/2 mm. Yeah now I plug that back into a partial fractions over here and seeing how one half, Let's have one over s. Plus one plus one half. I have one over S -1. So this is far enough for us to solve so get a little more space right now. Why Equal to 1/2? Has the universal laws transform of one over S plus one plus one half. I was in virtual applause transform of one over S. Uh I'm sorry S minus one. So now we recognize this as the applause transform of the exponential function. Yeah. You know it's going to come out to why is he going to one half, eat a negative T. Plus one half E. C. T. There you go.

Okay. So the information was given to this problem is that uh Why does the Prime 3? White Prime Plus two Y 0. Yeah and Y. Of zero. It was one and why prime and zero go zero. So if we begin to a little boss transforms, this will have S squared times little cloth transformed of. Why uh minus S. Y. Of zero. Yeah. Okay. Uh minus Y. Prime zero plus three outside of S. R. Fusions multiple design on the first try three S. Was that applause transform of? Why? Uh I only have minus wives. They will bring out the three sub minus three Y zero. Um and then close to I was a little applause transform of why Equals zero. So now I can plug in the values that we already know. So we'll have that S we're all transformed. Why? Okay minus X. Times Y zero is just at wiser. Was once, You know this is going to go zero Trina plus three S. Lockbox transform of Y mm hmm um -3 times four oz -3. And it's going to be plus to applause transform of why equals zero. It's not every time that's not attached to the boss transforms like native estimated three, you're gonna go into the right hand side and everything is attached to the cross transform, going to leave on the left hand side. In fact, a little false transfer between each term. So if you factor in the box boom, why? Why? Outside of S squared plus mm three s plus two Um has the Eagle two. Okay, as close brief. So now we want to isolate look lost right from the left hand side. You can divide both sides by this function here. So the last round strong. Why? Mhm. Eagle two S plus three over best squared um plus three S plus two. So now this um nominees easily factory. So we're gonna do um partial fraction decomposition on this. So I'm going to S plus three and then we have a three S. In A. Plus two. So we know this is going to be S. Plus one and I'm supposed to. Alright factors. It's now we're going to do the decomposition from A. Over. That's a plus one plus B. Very bad ones too. So we multiply this out and so equals X plus three. Uh A. S. Was to A. Okay, it was B. S. Must be Is equal to Ask Plus three. Yeah. Yeah. Alright, separated by terms around the S. Cosby S. So evil to earth around two. A. Plus would be yeah. Set equal to three. Yes. Yeah. Plus three. All right. So the vitality asses. I don't get that. Hey. Plus B. It is equal to one and then we'll bring that over here. She was counseling out of a plus B. Yeah. Should be cool. Which 1? So we're going to do is subtract this from the top because we only have one being each. We can cancel that for easily. So we're going to two A -1 as this is going to be a. And then B minus B is zero B. And then three miles one is a single too. So now we have that A is equal to two. So since we know that A plus musical to one, B has to be negative for me. So we have these values, you can put them back into uh decomposition over here. We have the two over it? S just one and we're gonna have a negative one. Yeah. Who were supposed to? Mm. Yeah. Mhm. All right. So um these two, the productions are pretty pretty easily recognizable as the applause transforms of the exponential functions. Right? Because you know that the applause transform of E. T. A. T. Is just equal. Yeah 21 over S my s a look. Mhm. So you know this. Uh huh. Two this will be times the boss transform of E. To A. T. And a. Here's our negative one. So it's the negative T. Um And I'll have minus and then it should be a little gloss transformed of each of the negative to T. Yeah. Yeah. So our answer is two ease and negative T minus. It's the negative too T.


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