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Use the method of Laplace transforms to solve the given initial value problem: Here, x' and y' denote differentiation with respect to tx' - 3x + 4y s...

Question

Use the method of Laplace transforms to solve the given initial value problem: Here, x' and y' denote differentiation with respect to tx' - 3x + 4y sin t 2x-y' -y cosx(0) = 0 Y(o) = 0Click the icon to view information on Laplace transformsx(t) y(t) = (Type exact answers in terms of e.)

Use the method of Laplace transforms to solve the given initial value problem: Here, x' and y' denote differentiation with respect to t x' - 3x + 4y sin t 2x-y' -y cos x(0) = 0 Y(o) = 0 Click the icon to view information on Laplace transforms x(t) y(t) = (Type exact answers in terms of e.)



Answers

Use the Laplace transform to solve the given initial-value problem. $y^{\prime \prime}-3 y^{\prime}+2 y=3 \cos t+\sin t, \quad y(0)=1, \quad y^{\prime}(0)=1$.

In the problem we have to what level does plus? Why does minus Y? Which is equal to sign treaty? Why F0 is equal to 0? Why does zero is equal to zero now? Had. So here we obtain the lap lots of why that is equal to three upon as a Squire plus nine in the to s minus one into S plus one. Now as this is a partial fraction so this can be solved here like this. So this is represented as three upon It is squared plus nine into two hours -1 into S plus one equals two. Airport S plus one plus They upon to S -1 -1 plus CS plus D. Upon esquire plus nine. No. Using hit and trial method we have solved obtained the result. That is a equal to -1 upon 10. Be equal to eight upon 37 C equals two -43.370 Day equals two 23 upon 3 70. Now, further this is served as lipless Universe -1 upon then into s plus one plus eight upon 37 into one. Upon Going to S -1 upon to plus -43. Bomb 3 70. Yes plus 23 upon 3 70. You want to buy? It's a squared plus nine. No this become -1 upon 10. With the power minus T Plus four upon 37. Heat Power T upon to -43. S upon 3 70 into esquire plus three square Plus 23 pond 3 70 in two it's a squared plus three squared. Now further this equals two -1 upon 10. It bar -1 plus four upon 37. A party upon to minus 43 upon 3 70 of course 30 Plus 23 upon 3 70. Sign 30. So this is the 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.

Which himself? Differential equation. Health is differential equation. Why? Double prime? Why crying plus two y You're going four terms, Uh, three t And this is what? The initial condition? Where wise dio zero. Yeah. Why? Prime of zero. The repository farm. You should be able to get their self square times. Why? As mass to be tired s two's. Why has why four divided by as many as three. Why, uh, is Holt four fire I that as a means to James Mass three. We could be too. Remain Is four by minus two. It was Y minus one. You okay? Bye. Yes, three. Then how do a positive version your years ago I t just cool till no four e que e to e to be okay.

In the problem they have Why double glass minus four Y. That is equal to three costea. Y. Of zero is equal to zero. And why does zero is called 20 So this is written as esquire lap lots of y. Now here this is the partial fraction form. So we have to obtain values of A B and C. So A is regionals. This is the value of A, B, C and D. So the value of A becomes 3.10. 3.10. So this is obtained by hidden trial method. Now we will find the hapless inverse. So lap listen words become 3.10. No I did. That is true upon turn into A. S plus two Plus three upon 10 And the S -2 plus -3.5 is upon As a Squire plus one. So this is region as 3.10, you'd power minus duty Plus three upon 10 years. The power to T -3 upon five Costea. So this is our Y f T. And this is the answer to the problem.


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