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(a) Using Laplace transforms, solve the following initial value problem:y" (t) 6y' (t) + 9y(t) = 3sinh(t) .y(0) = 1, %(0) = 1Solve the initial value probl...

Question

(a) Using Laplace transforms, solve the following initial value problem:y" (t) 6y' (t) + 9y(t) = 3sinh(t) .y(0) = 1, %(0) = 1Solve the initial value problem in (a) using MATLAB. and hence check your answer for (a).

(a) Using Laplace transforms, solve the following initial value problem: y" (t) 6y' (t) + 9y(t) = 3sinh(t) . y(0) = 1, %(0) = 1 Solve the initial value problem in (a) using MATLAB. and hence check your answer for (a).



Answers

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

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.

All right, so our problem here is why to the full problem when I was four or five equals zero, we're doing that Y zero which is one, why prime is equal zero, Y double prime zero equals negative two And white triple prime, zero equals zero. So we're gonna start this off by just taking a little loss transform each term sort of have S to the 4th, that's the boss transform of. Why minus s cube nope execute why zero? My ass squared white from zero minus S. Why double prime zero, where is s why triple four hours ago, All that -4 Little Postures from Fly Equals zero. So I'm gonna go through and look at the terms above and we have two terms why prime zero and what programs really? Both people Zero, this is going to go zero, I'm just gonna go zero. So here we have wives vehicles one, so you know this whole term is going to go to s cube. And over here we have a negative too. So you know this whole term is going to go to a plus to s It's not rewrite this after having simplify it a little bit. So I passed into the 4th plus stress from of y uh minus s cube plus to S Maya's for it's a loss transfer of why equals zero. Now we're going to separate out the terms that we do most times one of the austrians from y After out of S to the 4th, last four is all equal to s cube minus to us. Okay uh so now we're going to um get the transformed by itself, we're gonna have uh applause transform of why is equal to s cube Maya's to s over As to the 4th -4. Um So now we're gonna try to cancel some stuff out and get this to a form that we can work with so we're gonna factor and s out of the top, see how that goes S squared minus two. So on the bottom we can see that this as the fourth last four is a product of S squared minus four. I'm sorry escort advice to yeah times S squared plus two. Especially these to cancel out and we're left with um S over S. Squared plus. Alright so now where we go from here is um we're going to try to see if this imaginary of plus transfer. Let me know and it looks a lot like the little glass transform of a coastline function. Okay um But this needs to be a square. So we're gonna write like this S over S square plus Square root of two squares scenario, why is equal to the inverse laplace transform of S. Over S squared plus Square. Just two squared. So why is it equal to co sign A Squared of two T. There? It is

So here we have the function Y double prime minus Y tu minus two. Y prime plus two equals zero. We have the complete we have the I don't shoot sorry. We are the conditions that Y is 00 Y prime zero Equals one. So we start with our own Boston for do on the S where offerings from of why minus and it'll be S times wives 0 to 060 And then we move on my minus Y primatech form to yes. Why we're going to distribute the 22 This is zero anyway, so minus two times 500 Close to clubs to the next term April zero. Yeah. Sit right down so uh possible. So why after all these things that is connected to then move all other terms the other side. So one korean want isolate little classrooms from screen boston's realize you 21 over S where my S to S plus two man you too, which is equal to one over S minus one squared plus one, which correlates to the restaurants. And we already know that is for equals the inverse the flaws transformed of one over S -1 Squared Plus one. Because it's equal to the applause transform of E to the T shirts or a. Give me this term. Your honor be is this term in this term? So you can see this is going to eat to the one T. Time sign 212 as well. So I have that Y is equal to E to the T sign of T. Yes.

Okay. By nine. Minus 12. Ik teas in the Valerie. Why? Clearly wanted it with my primes hero. One using a Plautius equation. Toby s curtain. Why means best one which is equal 12. No way. So I asked to be rearranged two s where he is? Yes. Yeah. Time to amazing part, Frank. 10 0 years old. You're able to get made by or a kid. So then why t be able to buy he, uh, e two to the earliest two teas?


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