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Problem 2.points Take the Laplace transform of the IVP4 = 3 cos(kt); y(0)Use Y for the Laplace transform of y, (not Y (s))82 | k2and y(t)Note: You can earn partial ...

Question

Problem 2.points Take the Laplace transform of the IVP4 = 3 cos(kt); y(0)Use Y for the Laplace transform of y, (not Y (s))82 | k2and y(t)Note: You can earn partial credit on this problem;

Problem 2. points Take the Laplace transform of the IVP 4 = 3 cos(kt); y(0) Use Y for the Laplace transform of y, (not Y (s)) 82 | k2 and y(t) Note: You can earn partial credit on this problem;



Answers

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

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.

The first thing we're gonna do here is take the applause transform of our equation. Well, we do there, we're gonna get the applause transform of the second derivative plus four times of posturing from the first derivative plus four times of transformer. Why is equal to the boss transform Delta at T Myers for? And then we can use our rules or the class transforms of derivatives and delta functions to simplify this which we're gonna get s squared times, capital y minus s times y zero minus by prime of zero as our first turn plus four little times past times Why, minus y zero 1st 4 times capital Why is equal to plus transform this delta function are a is for Is that something you to the negative for? Yes. Now we're gonna plug in our initial values and factor out why capital I capital by times s squared plus for s plus four minus asked Times y zero y zero, we're told is just one This is minus s minus two. Help us four we ever caused or four times and negative one was negative for is equal to you. The negative for s starting for capital while will give us eat in the negative for s plus s plus six all divided by X squared Plus for s plus four. We can factor the denominator here beating e to the negative for s plus s plus six All over s plus two all squared. And now we're going to separate this inju three fractions, one with either the negative for s I was one over s plus two all squared plus s over s plus two All squared plus six are s plus two well, square and we're going to use partial fraction decomposition on the center fraction. So we're gonna have s over s plus two all square and want to write this as a over s plus two for being over s plus two squared. Let me carry through The denominator we're gonna get asked is equal to a Times X plus two Well, plus, being this is equal to a s plus to a plus, B is equal to s. This is s plus some just zero. So therefore, a has two people. The one been to a plus being has to be equal to zero, but a is one. So be pressed medical the negative too. Therefore, this fraction will be won over s plus tune minus two over s plus two all squared. So therefore, our capital why will be eating to the negative for yes times one over s plus two, All square plus one over as opposed to you. Minus two over s plus two. All squared plus six over est was too elsewhere. You see that? These two fractions over here just going to add to four over S plus two, most were. And now we have three fractions of which we're going to take the inverse of loss transform. It's between the inverse of prostitutes from the capital. Whatever is gonna get a why being able to inverse a farce transforming needed negative for s times one over s plus two all squared Plus in verse applause Transform of one over s plus two plus in versus loss. Transform of carrying the four outside one over s bus to all squared. And we're gonna start with the second function here, and we're going to use the I believe for shipping there. She's in the way of the posture. Answer with E to the negative A s Yes, time some function on T. This will be equal. Teoh, our f That s minus a. And when you can see that we have that here, This is just one over s minus negative too. So therefore, this will be you to the negative to s times the universal class transform of one over s which is just one. And we have that same thing over here. This is just one over s minus negative, too. Most weird to inverse applause. Transformer this well, the need to the negative to us times the inverse applause transform of one over X squared. We can use the rule that the boss transformer T raised a cell manager and vicious n factorial over STV n plus one and one over X squared is nothing other than one factorial over s to the one plus ones or an is one here. So this is just teeth. This second term will Then we need to the negative to us times T and finally this fourth term for this first term, we're going to have to use the second shipping there as well, which says that if we have the hearts transform of a heavy side function times a function at T minus A. This is a sequel that you need to the A s. You know, the negative A s times lacrosse, transform of that function f So we use the second shifting theorem. First, we know that this will have to be the heavy side. At four, 14 times the inverse I'll toss transform of one over s plus two All squared Evaluated a T minus four. But we just figured out what the cost transformer of one over X Plus two squared is over here. So this will be the heavy side for t times e to the negative, too. Your team no gets you teen. We have to subtract four from t times t minus forward. Therefore, gathering this term this term in this term Now, why will be the heavy side before of tea kind of you to the negative to 15 years for all times T minus four. Plus factoring out you the negative to t from the 2nd 2 terms, they 14 plus one

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.

To start here. We're gonna take me flash transform of Ah, different equation here, and we do. So we're gonna get the Lacoste transform of the second derivative minus three times of applause transform. The first year of you minus four times have looked washings from why being with a former terms of across transform of e to the negative team and we can use our rules for the plus transforms to simplify this, which means that this would be a square, intense capital y minus s times by zero minus y prime was zero as a first term minus three times as 10th capital I minus y zero minus four times capital. Why, equal to four times a plus Transform even the negative team based on this 21 over s Respondents Sensei's negative one. Now we're gonna solve for capital. Why? By factoring it out from the terms that we have year, which will give us s squared minus three s minus four. All times capital I turns, we have left our a minus s minus one and plus three. Vehicle to four over s plus one. Now we're gonna solve the capital lie here, So we're gonna get capitalize equal to four over s plus one times s squared, minus three ass minus four Plus aspirin is too all divided by s squared minus three ass minus four. And we can factor this denominator into best minus four. And thats plus one that goes with this over here too. And we can multiply this numerator by a factor of s plus one over s plus one to get a common denominator and merge the two fractions. And when we do so we're gonna get that This is as squared minus s plus two, all divided by s plus one squared times asked minus four. Now we're going to use partial fraction decomposition on this. So we're gonna write s squared minus s plus two over s plus one square terms s mines for as one for action. They are restless. One plus some being over s plus one squared. Plus some seeing over s minus four or any carry through. That's not denominator to get that s squared. Minus s plus two is equal to hang. Times s plus one turns s minus four plus being time Just minus four. First seen time just was born ball squared. And now we're going to select some values for s to make this easier. So S is equal to negative one. Bringing the negative one squared is one minus negative One is trust one. It's nice to zeros you any times negative one plus one is zero plus being times negative. One minus four is negative. Five plus C time zero sh zero So for the left hand side of the two plus one is three plus one is four divided by negative five. So be is negative for fifths. If it's yours s do you floor? Forget four squared is 16 minus four plus two is the will of 14 giggles you any times? Four months for zeros, Mrs Zero plus B time +00 But see, any times four plus one, which is five all squared. Therefore see is 14 over 25. Now we can pick any old s such big zero for ease. So left side we're getting it too. Equals you A Since one times negative floor plus B which is negative. 4/5 turns negative for plus seems just 14/25 times one we're then gonna solve for a so a is to? Well, any times negative floor. Okay, is to minus 16/5, minus 14/25. I'm gonna find a common denominator. 25 will be 50 over 20. Size minus 80 over 25 minus 14/25. Well, give us negative. 44/25. It's therefore a is 11 over 20 size. It's there for our fraction. Partial fraction. Decomposition is 11/25 turns one over X plus one minus 4/5 hands one over s plus one squared. Plus 14/25. That was one of the S minus four. And this is equals. Who are capital? Why? We're gonna want take the inverse laplace transform everything. So if we take the interest, boss, transform our why? I will be able to 11/25 times the inverse LaPlace transform of one over X plus one minus four. Fits turns angry supplies transform of one over. Yes, plus one squared plus 14/25 in verse. The cross transform of one over s minus four. And now we're gonna use our rules. Other rules for the class transforms up here, and I'm going to start with the first and third terms since there just follow as eat as one over s minus a are they will be negative one. So this will be eat the negative team. And this 3rd 1 Inverse LaPlace Transform for a is four. So it should be any to the 14. This one right here If we use the first shifting theorem, will see that this is e true, Mazzini. Negative t times inverse LaPlace. Transform of one of her s squared. Just tea. So therefore, why is equal to 11/25 times in the negative? T minus 4/5 times T turns either the negativity plus 14/25 King to the 14th.


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