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Consider the following initial-value problem: Sy' 8e4t Y(0) = 1,Y0) -1Find Kf(t)} , for f(t) 8e4t 4e (Write your answer as function of 5.)Af(t))Use the Laplace...

Question

Consider the following initial-value problem: Sy' 8e4t Y(0) = 1,Y0) -1Find Kf(t)} , for f(t) 8e4t 4e (Write your answer as function of 5.)Af(t))Use the Laplace transform to solve the given initial-value problem:y(t)

Consider the following initial-value problem: Sy' 8e4t Y(0) = 1,Y0) -1 Find Kf(t)} , for f(t) 8e4t 4e (Write your answer as function of 5.) Af(t)) Use the Laplace transform to solve the given initial-value problem: y(t)



Answers

Use the Laplace transform to solve the given initial-value problem. $y^{\prime}+y=5 e^{t} \sin t, \quad y(0)=1$.

So for Why? Why you keep it, you five. And why weren't three year old hold Teoh One? So that'll be rearranged This plus tree Why, right? I notice one when you came Why? Who follows before? Why has equal to by s Square one Waas wanted to buy by this queer was four. Why I fine to t plus.

In the problem we have Viral Illness -4. Mhm minus Plus five. White at Equal zero. Why have zero is equal to one? And why does zero is equal to two? Every day's a Squire. Lap plus white minus yes. Why is 0-? Why does zero minus forward into earth? Annapolis y minus Y zero plus five Last last. My that equals zero. So it is is required. Yeah plus Y- is -2 minus four. Is that plus one? Plus 4? Plus five. Hapless Way equals zero. So lap splice hawaii indu is a Squire minus forest Plus five. That equals S -2. Or lap list will become is minus to a bomb. Yes -2 whole squared plus one. So we have Y. F. D. Equal that blessing was all Is -2 upon is -2 Whole Square Plus one. So this is recognized you. The power minus sorry to the power to t costing. So we have we left becomes you have the power to D costing as our answer.

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.

In the problem we have four wide or less Plus four. White as my plus five. Why that is zero Or Y. of zero is equal to. Wanna. Why does zero is equal to -1 upon to? So we have four into it is quite lap lots of hawaii minus is Y zero minus minus zero plus four into S. Lab plus Y minus Y zero plus five plus. Why? That equals zero. Our father we have for esquire lap plus white minus photos plus two plus forest, lap plus white minus four plus five lap plus Y. That is each call to zero. Hence we have lab plus Y. Into for esquire plus forest plus five. That equals for S. Plus two. So laplace of Y. Is equal to forest plus two upon To s plus one Whole Square Plus two. Police question. Now further we have Y. F. D. Equal laplace in brasov. Forest Plus two upon To S Plus one. All is quiet plus two square. So further we have Y. F. D. Equal to to into into s. Plus one upon to bon forward into S. Plus gonna 0.2 whole square plus four. This is equal to S. Plus half. He wanted to buy. Yes plus on upon two whole square plus one. So this is our lab plus in verse and this is our last lesson worse. Therefore, why have they become? You're the power minus T. Upon to costume. So this is the answer.


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