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Use Laplace transforms to solve the initial boundary value problem UII, T > 0, t>0, U1(0,t) - u(0,t) =0, t>0, u(T,0) = Uo, T > 0. Interpret= this model ...

Question

Use Laplace transforms to solve the initial boundary value problem UII, T > 0, t>0, U1(0,t) - u(0,t) =0, t>0, u(T,0) = Uo, T > 0. Interpret= this model physically in the context of heat flow.

Use Laplace transforms to solve the initial boundary value problem UII, T > 0, t>0, U1(0,t) - u(0,t) =0, t>0, u(T,0) = Uo, T > 0. Interpret= this model physically in the context of heat flow.



Answers

Use the Laplace transform to solve the given initial-value problem. $$y^{\prime \prime}+2 y^{\prime}+y=\delta(t-4), \quad y(0)=0, \quad y^{\prime}(0)=0$$

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 Y. Two prime minus Y. Two Y. Prime plus two. Y equals coastline. T. We know the wise vehicles one. Why prime +00. So starting the boss transforms. Okay square the plus transform why minus S. 10 Last Y. Prime of zero. My S two times S a glass of wine. The story that -2. Is this gonna be a plus two 10? It was too plus restaurant. Why equal to the little applause transform? A co sign of T. Which we know is able to pass over S. Squared. What's one? So now I have we got sympathize out so we know that This term is gonna go to zero in this term is going to go to S. And this term is going to go to two S. Squared plus transforming why My S. -2 s. Fox. Transfer of why? Plus two plus to Austria's or why equals S. Over S. Squared plus one. Now we're going to keep going civilization escort advice to as supposed to 02 S. S squared plus one Plus S -2. So we're gonna try to get this, get these two numbers into the new winners by doing that. We just multiply them by the denominator. I hope that first term. So it's going to be the S. Plus S. Times S. Squared plus one. Which is going to be S. Cute plus S. And the negative two times the same things as a reminder to s where mines to all over um S squared plus one. Yeah. Okay so now we're gonna bring this down here and we're going to do uh partial fraction he's elimination. I'm sorry. We're gonna let's get a step Estonia isolate this loss transform. So we have um S plus sq dot Mt. Rearrange this. Why did you have s cubed admires to S squared Plus to S -2 all over. Oh you do all over S squared plus one. Multiplied by one over. S where my ass to S plus two. So we're just gonna Eagle two. I'm gonna s cube last two. X. Squared Plus to S -2. All over. I squared plus one and S squared my ass to S plus two. Thanks. So now we're going to do partial fraction decomposition. I'm gonna get A. S. Mhm. Plus B. Over our first thing. So there S squared plus one plus CS plus D. Over S squared minus two S plus two. It's not really multiply each numerator by the opposite denominator. And for our first one we are going to get um Mhm. We are going to get uh sorry uh Here we go. A. S cute um minus to S squared plus B. S squared plus two S minus two Bs plus to be. And for our second side running it plus CS cube plus Ds squared plus CS. It helps, you know um bryson mm plus CS plus D. And all of that is going to be equal to our new mayor. Before we write this we can all have on the same line. Um A. S cuse minus two. A. S squared plus B. S. The square that plus to us. Um Two. Yeah. Yeah. Yeah. All equal to a numerator which was if we look up here it was S cuse my last two S squared plus two. S last two. All right. So I'm going to separate out our life terms will have s cube. So first we have our Charles. I haven't sq the numbers are a. S. Cubans. He has cube. Then we have our squares which is negative two X. Squared. B. S. Squared and Ds. Where we have our just plain asses which are two s native to Bs CS. And that will have to be nd and their corresponding terms will be here here here and here. That helps off. So um A. S. Cube plus CS cube. It's legal to ask. Cute. We'll have negative two S squared plus B. S squared plus DS. Where it is equal to -2 s. Where um And then we'll have ah to a S minus two Bs plus CS equals two S. Um And then for our last term will have to be yeah. Plus D. is equal to -2. Alright so now we're gonna start we're going down the line and canceling on rs terms so we can divide this all by it. S is here. But everything I squared here. But everything my ass here. So now I have a plus C. Is equal to one, negative to a plus B plus D. C. Goes to negative two to a class to be Plus C is equal to two. So now we're gonna have to figure out a way that we can swap in terms in these equations and figure out values from afar. Mhm numerous our values here. So I mean Looking across these two middle equations was in our center here um are common terms or A. And B. So I'm going to try to express C. And D. In terms of A. And B. So over here I'm going to press C. As one minus A. Are far right side. Follow me over here and have no interest D. As negative two minus to be someone. Uh enter these terms into these equations here. I guess swap those lines. I want to bring these values over here. Mhm mhm substitute I wasn't right here substitute and continue itself and simplify. So this is gonna be negative to a right here in the middle plus B. And then I mean D. Is negative. So it's a B plus negative to my ass. To be. You go to 92. All right to a one has to be plus one. Might say eagles too. So now this is going to be negative to a A plus B. With a nice basis, we might ask one B -2 Equals -2. So this comes down to negative to a Must be equals negative or evil zero. Because I'm gonna add that tour there. Mhm. Turn over here we're gonna have two A -1. A. So this is going to be okay. Yeah minus to B plus one equals two. So it has attracted one from outside around the A minus to B equals one. I have these two equations are both in terms of uh A. And A. Only in terms of A. And B. So I'm gonna bring us another person over here so I have negative to A. My speed equals zero. And to cancel that term I see that uh this is A. Is a negative too. So always whole top equation by two and add them together. So I have zero A. Because to a -2-a-0. It's gonna be negative for b minus peace of mind. Negative five B Equals and this is going to be two times 1. How serious this is. Oh sorry. Um Our cameras, there you go. Change. It's going to be to be equal to negative two fists. They are first time. All right, so we're bringing over here yeah the negative to A. Is equal to uh sorry let's do this more time there to a plus 2/5. So go to zero. That means that negative to a ceo to the negative two fists which means that A. Is equal to 2/10. As that term is now we're gonna use that value to solve for R. C. Value. and so see is equal to one minus 2/10. It was A. C. Is equal to 8/10. And then we're gonna solve our final values is D. Two. Native to minus two Times Native two or 5. Which is going to be able to bring everything the fifth to make this easier for negative to negative 10 over five plus or over five. Right? Um Yeah. Uh Yeah okay so um this is not going to be D. Is equal to negative mm 6 50. All right Yeah. Mhm. Now we can um double check these values. We will by going let's try it for deep. What's going on? So next to A that's gonna be um negative two fists minus two fists minus 6/5 Equals -2. Let's see So it's gonna be negative to Mars -2 -96 consume negative 10 Or five. She goes -2. So these all work. So that leaves you with that sees us correct because everything else is correct. Alright so now we're gonna swap these back into our um yeah are first rations of here and we're going to get that so a. Is too tense. Try and get To over 10 s Plus and be with -2 fists 1 -2 fists was there over S squared plus one Plus & C. was 8/10. Right. Yeah plus 8/10 S. Indeed is -6/5. Yeah So -6 or five mm over S squared minus two S plus two. All right. So, I'm gonna erase all this work down here so we can uh I would use your space to work. We've seen all this friday. You'll need to see this anymore. Do do do race. Yeah, be thorough. Alright. Still more most. Yeah. Well, I was getting almost all of it. Probably good. No. Mm mm. Uh I got there. Yeah, scroll back up and right, okay. All right, let's get back to this problem. All right. So now we've done our part of fractions. Now our job is to uh separate us out into a form that we can recognize the little plus transfer. I'm gonna tackle first one term at the time. So our first term here is going to be this one. So first of all, let's separate this out. So we're gonna have two or 10 outside of S over S squared plus one minus 2/5 or one over S squared plus one. All right. So that salsa for this side. So these are these are now forms that we recognize as applause transform. So that's good for that side. So now we're gonna tackle the right hand side and we'll combine them and say all right, mhm. This side here, I'll separate this out as well. Okay. Uh Let's look at this. So if you look at right here, we see that our numerator is going to be ah we're able to see the numerator is made up of um a certain polynomial based on the S one S two S which is going to be s minus one squared and then we'll have plus one because s minus one squared gives us s square advice to S plus one. Well, plus two appears we're gonna need another one down here. All right, so now we're gonna work just with this numerator here because we know that we're gonna need it M. S. My last one up top. So you know that's multiplied by an 8/10. Aren't going to simplify it down to 4/5. So 4/5 It was me, 4/5 minus 4/5. So to get to 6 50 we're gonna have to subtract another 2/5. All right, so that's gonna be a new new marina. Already have s minus one squared plus one down here. All right, so we're gonna separate the sale just like we did to the left hand side, We're gonna get 4/5 Outside S -1 Over S -1 Squared plus one minus 2/5 outside of one over s bias one squared plus one. All right, so now we can see that this is going to why is equal to the universal applause transform of Uh sorry two or 10 um To s squared plus one minus 2/5 inverse applause transform of one over S squared plus one plus 4/5. Universal applause transform of S -1 over pass minus one squared plus one last 2/5 University plus transform of one over s minus one squared plus one. All right. So I can see that. Why is going to be equal to two Over 10 times so far down 1 5th This final step, so 1/5. Um You can see this, right? This is as you can see this is the form for a co sign functions as we co sign of t minus 2/5 sign of T. It's a random wine here in the middle. All right. Um Plus 4/5. Eat it. He co sign of T last 2/5. Um Eat it. He sign of T. That is your answer to this long problem.

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.


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