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Use the Laplace transform solve the follwing initial value problem:10y + 24y = 0, y(o) = -4, % (0) =(1) First; using Y for the Laplace transform of ylt), ie. Y = C(...

Question

Use the Laplace transform solve the follwing initial value problem:10y + 24y = 0, y(o) = -4, % (0) =(1) First; using Y for the Laplace transform of ylt), ie. Y = C(y(t)) , find the equation You get by taking the Laplace transform of the differential equation to obtain(2) Next solve for Y(3) Now write the above answerits partial fraction form, Y =(NOTE: the order that You enter your answers matter 50 You must order your terms 50 that the first corresponds -2 < 1)and the second to whereAlso not

Use the Laplace transform solve the follwing initial value problem: 10y + 24y = 0, y(o) = -4, % (0) = (1) First; using Y for the Laplace transform of ylt), ie. Y = C(y(t)) , find the equation You get by taking the Laplace transform of the differential equation to obtain (2) Next solve for Y (3) Now write the above answer its partial fraction form, Y = (NOTE: the order that You enter your answers matter 50 You must order your terms 50 that the first corresponds -2 < 1) and the second to where Also note example that (4) Finally apply the inverse Laplace transform find y(t) y(t)



Answers

Use the Laplace transform to solve the given initial-value problem. $y^{\prime \prime}+4 y=0, \quad y(0)=5, \quad y^{\prime}(0)=1$

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.

Why's he here? My fine here. Applause for male Pretty angel s squared times Why? My simple via wire equal to s plus high I were Teoh be which can single fight to be to just one who's two time are You may be here.

Trying to do the posturing form. Why Double prime? Why double place me prime pushed to He's going for war for a wise hero. Hong's Europe with I Prime born, not one. Why prime zero? He calls. It's a little beat doing posturing form square minutes three s Who's to why has multiple my likeness minus one. This one he pulls four divided by passed away Simplifying. Why is for I I That's raise one Delta pi pi as still and I get a parking back Fine divided by and it's one to us To this three. It s a little passing personally, whitey equal to me e e t plus.

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


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