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Use the Laplace transform to solve the following initial value problem_y" + 4y = f (t) , y (0) = 2,y' (0) =3wheret < 2 f (t) = {2 t2 2 Procede using th...

Question

Use the Laplace transform to solve the following initial value problem_y" + 4y = f (t) , y (0) = 2,y' (0) =3wheret < 2 f (t) = {2 t2 2 Procede using the following steps_ Find the Laplace transform of f (t). If Y is the Laplace transform of y; find the Laplace transform of the left side of the differential equation. Solve for Y as a function of $. Find the inverse transform of Y to solve the differential equation Write your final answer as piecewised defined function:

Use the Laplace transform to solve the following initial value problem_ y" + 4y = f (t) , y (0) = 2,y' (0) =3 where t < 2 f (t) = {2 t2 2 Procede using the following steps_ Find the Laplace transform of f (t). If Y is the Laplace transform of y; find the Laplace transform of the left side of the differential equation. Solve for Y as a function of $. Find the inverse transform of Y to solve the differential equation Write your final answer as piecewised defined function:



Answers

Use the Laplace transform to solve the given initial-value problem. $y^{\prime \prime}-3 y^{\prime}+2 y=4, \quad y(0)=0, \quad y^{\prime}(0)=1$.

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

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.

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.

White year spin the whole differential equation. Which is why prime my crime was to Why, for Pete using the utility neutral boundary of y zero u calling one possible plus transform the asked him apply as please one post times. Why? Because he goes for it by weird and very arranging it. Why, as he four plus s as Bastille in school. Que these wondering why e to to means she t res one? Yeah.


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