5

Point) Consider the following initia value problem:y" _ %y' 24y sin(3t)9(0) = -4, y (0) = 4Using Y for the aplace transform of y(t) ie_Y = C{ylt)}. find ...

Question

Point) Consider the following initia value problem:y" _ %y' 24y sin(3t)9(0) = -4, y (0) = 4Using Y for the aplace transform of y(t) ie_Y = C{ylt)}. find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)

point) Consider the following initia value problem: y" _ %y' 24y sin(3t) 9(0) = -4, y (0) = 4 Using Y for the aplace transform of y(t) ie_Y = C{ylt)}. find the equation you get by taking the Laplace transform of the differential equation and solve for Y(s)



Answers

Solve the given differential equations by Laplace transforms. The function is subject to the given conditions. $$4 y^{\prime \prime}+4 y^{\prime}+y=0, y(0)=1, y^{\prime}(0)=0$$

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.

In the problem we have Y double DAS plus four Y equals zero. Now Y. Of zero is equal to zero. And why does zero is equal to wanna? So it is esquire lap lots of Y minus is Y. Of zero minus virus. Zero plus four. Lap lots of why that is equal to zero. So it is S squared plus four lap lists of why that equals one. So like a place of y equal to one upon esquire plus to the power of two. So Y. F. T become lifeless universe. One upon as a Squire. Plus to the power to this is equal to half lap. Listen verse two upon, it's a squared plus two squared. This is equal to half signed to D. This is our Y. F. T. And this is the answer.

So for this one we have uh Y double prime minus four, Y plus four zero. We know that Y zero equals one and Y prime 0 to 1. So that was taking place from from all of this from this we're lost. Why? Uh general My as Wy 0 uh minus for s classrooms of why plan for why zero plus four plus four, Y equals zero. Um You know this goes to one, this goes to one and this goes to once we have S square classmates or why My S Vice one minus four S class transfer of y minus four plus four, wash my soul of life user. So now we're going to bring all the terms that aren't attached to the pasta from over to our right over to our right side. Yeah, all that. And then all the terms that are attached to the boss transforms, we're going to factor out. Sure has, we're going to factor out the laplace transform from these terms S squared last four S plus floor. It's equal to S plus five which I got eyes from adding the pass to this side of having the one and four. So as plus one plus four X plus five. Then we want to get the conventional loans, we're gonna go get go S. Plus five All Over. S where blast or s plus four Eagle two S plus five all over S minus two square. Uh So now with this we're gonna go uh into partial fractions. So now that A. Or S. Mine is too plus B. Over S minus two squared. And to get the same denominator. All you do is do A times S minus two Must be evil. That's plus five. It's going to be A. S minus to A. Plus B equals S. Plus. Uh Yes every amount into life terms. So have A. S. Equals S. And negative. Sorry about that. Negative two A. Must be eagles five. So cancel these out for me. A. is equal to one. So this is negative two Plus B equals five. So b equals seven. Right? Those are two terms. And then I'm just gonna put them back into a partial partial fractions. I mean everyone get that one over. That's my two plus seven over as far as to square. That's now, you know the last one from of why it's equal to the universal class transforms one over s minus tube it And the university blossoms one of bring the seven out. Mhm. Yeah. Yeah. Every last one of seven over. Uh huh Transform Universe lost transform of one over S minus two squared. So this is enough for us to go off of to finish this problem. I'm going to take the inverse washing everything we get why it's equal to. And then we're going to see that the first one we recognize as a form of little plastic from E. To the They have to be a three e. 2 t. Plus seven times we recognize that one. Sorry as uh Yeah. Uh Our local transform of T. To T. To the end of E. To the T. Where um This is our value attached to the T. Value attached to the experts. The order number Just give me seven t. 2 e. two T. That is our answer to this problem.

Okay so this problem if y double time last four Y. Prime plus four Y equals zero. Was not white wives miracles one. My problem is you're already wants to start a national start to the bathroom on this. Where? Why my ass s might as well. I mean I can do this because we have these facts here. Uh My last 41 Y distribute the four. Just four. We not only times or once with a slew of 4-plus 4. Yeah. Why he goes there? Alright so now we're gonna move all the terms not attached to a little plus transfer over to the right and for all the ones that are we're going to factor out the applause transform collecting pregnancies. Why is it to her? At times? S very last for us. Close for he was S plus five because that's these terms I end up on the right side. Now we're going to get a little foster restaurant alone during this uh S Plus five over S squared minus four. S plus four equal to S plus five. All over S -2 Squared. Uh So it is we're going to do partial partial fraction decomposition. Okay. Over As -2 Plus B over S -2 Squared. And to get the same thing on there always to do with multiple a. S -2 Serena AS -2. Philosophy to be able to s plus five just go to K. S minus to A plus B equals as plus five. Never have uh S equals s separating out like terrorism. I used to a plus B equals five. So for this we can always see that a close one. That's what we're going to use to philosophy 25. So be equal seven. I have our enumerators trying to get us back to the partial reactions we had earlier up here. Okay, I have that one over. That's nice too. It's plus seven over s last two squared. It's not gonna bring it back to the posthumous for the plasterers. From why? Sure. Let's just do this. Just my good. Mhm. Why is equal to the inverse applause transform one over X -2 Plus seven times injuries cross transform of on over sms to squared. And now we recognize these uh transforms or invincibles transforms so we can make this Y equals E. To the to T Plus 70 e. 2 to T. And that is our answer. Mm


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