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Question 1 (Ex 24 of Section 5.2) . Find the Laplace transform of the function that would solve the following IVP. (You do not have to, but are welcome to try to, f...

Question

Question 1 (Ex 24 of Section 5.2) . Find the Laplace transform of the function that would solve the following IVP. (You do not have to, but are welcome to try to, find the inverse Laplace:_ y"' +y + 2y cos 2t + sin 3t, y(0) = -1,%(0) = 1 .

Question 1 (Ex 24 of Section 5.2) . Find the Laplace transform of the function that would solve the following IVP. (You do not have to, but are welcome to try to, find the inverse Laplace:_ y"' +y + 2y cos 2t + sin 3t, y(0) = -1,%(0) = 1 .



Answers

Find the Laplace transform $Y(s)=\mathcal{L}[y]$ of the solution of the given initial value problem. A method of determining the inverse transform is developed in Section $6.3 .$ $$ y^{\prime \prime}+4 y=\left\{\begin{array}{ll}{1,} & {0 \leq t<\pi,} \\ {0,} & {\pi \leq t<\infty}\end{array} \quad y(0)=1, \quad y^{\prime}(0)=0\right. $$

All right? So the problem here is Y double prime plus four. Y. Is equal to the step function which says it is eagle. T. When T. is between zero and 1 or less than one. And as you have one when T. Is either one or less than anything in the infinity. So our first step again is to visualize this step function. That was right. An equation for it. And we know that from zero to one here. I'll do the uh the actual the line in different colours here, cuz Mhm. So we know that from 0 to 1. It's just evil to T. And At one at 1 Or just equal to one. Yeah. The rest of this crap. So that's why we have so let's see. So we know that's going to be this is equal to t minus. It's going to be you one of T. The only figure out what it's multiplied by making one. And so for t to always be uh to what you need to subtract from T to make sure it's always gonna be one Is T. itself that makes zero out of one. So that makes it. So this is consistently one that was rewrite. Our original equation Should be y double prime plus four y. Eagle two T minus T plus one. You have one and you can write you one of T. Or you have one into the matter. I'm gonna start excluding the T. Just um safe space. It's not a little pause transform. We're gonna s where after this world Y. And then up here we can see that Y. 00 and 100. So we can just exclude where normally we have S. Y. And so on. We're going to exclude that because you know it's all going to zero. So I said recent right? That's weird Blossoms from Y Plus four. Last chance for fly um to the la plaza transform of T -2 applause transform of G plus one. You have one. Uh huh. On his left hand side were in fact that little floss transform S squared plus four on this side we know that the applause transform of tea is joy is just going to be um Yeah so here we have you know that the applause transform of t to whatever power it is. Let's call it and just eagle to end factorial over As to the end Plus one. So right now we have t to the one power. So this is the one vehicle to one over S squared. So now our formula for uh course our family for the function where we have uh a step function times uh another function. So in this case is T. Plus one. What you can do is take the step function out. Yeah and then change the the function left over to um reflect that. And I will show you this form in the second. So basically what is is that if you have a little blocks transform of a step function it's called a times another area also called ti yeah mr T uh minus a. All you gotta do is take it out and make this um you make it E. U. Use the exponential function. This will become E. To the negative. I'm sorry how long they were about do my parents? Okay so all you gotta do is take this out and make it E. To the negative A. This has become you too eat a negative A. S technical flaws transform of. And what you'll have remaining is you add that a back so you just have the T. S. Over. So in our case we have t. Plus one. Yeah you have one. So we know. So in this case we could see that always to do is add the one back on there. So it's gonna be minus. Eat a negative S because it's usually you want Loss Transform. And since we're adding one T. Plus to this case we can simplify this further past transform of why? S squared plus four what equals one over X squared minus E. To the negative S. Okay. Actually this place is not that we can we can figure it out without doing that. I'm going to erase to safely space race these things there. Alright so now start taking the class transfer of these things so we're gonna leave the left hand side from now. Watching from Y. S squared plus four equals one over S. Where S. E. To negative S. Um multiplied by And then the boston tea is again that we want to rest squared. You know plus runs on the tube is going to be too over S. That's almost for that one over X squared minus. He's a negative S over S. Squared plus or homicide. Another minus. Um To a E. Mhm. Uh huh. To negative S over S. I'm gonna do this again. Why? That's where plus four next we're gonna get those all the same denominator. So we already have these are the same the first two on the same night when you multiply this term by another. S just come out to one minus E. To negative s minus two. S. Even negative S all over X squared. The final step is getting rid of the this on the left hand side this boss transfer of Y equal to one last E. To negative s minus two S. E. To the negative S All over X squared multiplied by one over S squared plus four. Right. Sure. It it's not always to do is distribute. That's where our originals on there. Over to what we're multiplying by L. Y. Possible from Y. Legal too One last each a negative S -2 s. Each negative s. All over as to the 4th plus for us. And that is your last chance

Why Prime Minister Wide five times she Why has life here are one. And is it for why you Mais s where 60? Yes, right. You just gave me kill as to s means squared course for going to kid que thank you.

All right. So our problem here is wise old prime plus two. I promptly fired White zero. Were given that Y zero. It was too with my prime zero, negative one started the applause transform the S squared why uh minus? It's going to be s times Y zero which means to me to us and it's going to be minus why privacy is negative one plus one. Yeah, plus two. As why are we going to be minus two times Y zero in minus for just five classrooms from why eagles here. Alright, separate them out. S square plus two S plus five, remove all the other terms to the right hand side. This can be to S minus one plus four, so plus three. Not to get the questions were alone will do pollsters from Y to go to to S Plus three all over S squared plus two S plus five. Okay. And now from here it kind of looks like we can work this into a form that we recognize and try to make this into the past transform of the exponential function as a senior sort of function. So first we're on now here around two F plus three all over. And then if you look here, you see the S square plus to us, you know, that's the beginning to the function as plus one squared, which we know will come out to s squared plus two S plus one. But we need to plus five. We're going to add another four. Now if you carry on. But the bombs we're gonna have to outside of S plus one. You need that up top for this for this form we're going for. Uh And then we need to get a plus three but it's going to get us to s plus to add another one. Here There you have s. Plus one squared plus two square. So I'm gonna write this two squared because in a loss transform for next bunch of function times as high as soil function. Um This term here is has to be a square. So now we're gonna separate this out. We'll get that too. Times S-plus one all over. S plus one squared plus two squared plus one over S plus one squared plus to start. All right? But here on the on this side here we know that these terms need to match. So rewrite this to S plus one over S plus one squared plus two squared Plus we'll take out 1/2 to make that equal. Well we have to up here S plus one squared plus two squared. So now we can solve especially why is he going to the applause transform of took two out. So I forgot uh S. Plus one over S. Plus one squared. Kind of messed it up. Yeah. Yes plus one squared plus two squared. Uh Plus one half. Universal boss transformed of two over S plus one squared plus two straight. Uh What we know from Y. Equals to eat a negative T. Uh co sign of two T. Plus one half E. To the negative T. Sign of two T. That is the answer to that bro.


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