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Use the Laplace transform to solve the Initial Value problems of the following: dy Q9 a) 4y Ssin3t ,y' (0) = 1 dtb)y" T Sy + 4y = e-2t 'y(0) = 0,y&#x...

Question

Use the Laplace transform to solve the Initial Value problems of the following: dy Q9 a) 4y Ssin3t ,y' (0) = 1 dtb)y" T Sy + 4y = e-2t 'y(0) = 0,y' (0) = 3

Use the Laplace transform to solve the Initial Value problems of the following: dy Q9 a) 4y Ssin3t ,y' (0) = 1 dt b)y" T Sy + 4y = e-2t 'y(0) = 0,y' (0) = 3



Answers

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

Okay. So the information was given to this problem is that uh Why does the Prime 3? White Prime Plus two Y 0. Yeah and Y. Of zero. It was one and why prime and zero go zero. So if we begin to a little boss transforms, this will have S squared times little cloth transformed of. Why uh minus S. Y. Of zero. Yeah. Okay. Uh minus Y. Prime zero plus three outside of S. R. Fusions multiple design on the first try three S. Was that applause transform of? Why? Uh I only have minus wives. They will bring out the three sub minus three Y zero. Um and then close to I was a little applause transform of why Equals zero. So now I can plug in the values that we already know. So we'll have that S we're all transformed. Why? Okay minus X. Times Y zero is just at wiser. Was once, You know this is going to go zero Trina plus three S. Lockbox transform of Y mm hmm um -3 times four oz -3. And it's going to be plus to applause transform of why equals zero. It's not every time that's not attached to the boss transforms like native estimated three, you're gonna go into the right hand side and everything is attached to the cross transform, going to leave on the left hand side. In fact, a little false transfer between each term. So if you factor in the box boom, why? Why? Outside of S squared plus mm three s plus two Um has the Eagle two. Okay, as close brief. So now we want to isolate look lost right from the left hand side. You can divide both sides by this function here. So the last round strong. Why? Mhm. Eagle two S plus three over best squared um plus three S plus two. So now this um nominees easily factory. So we're gonna do um partial fraction decomposition on this. So I'm going to S plus three and then we have a three S. In A. Plus two. So we know this is going to be S. Plus one and I'm supposed to. Alright factors. It's now we're going to do the decomposition from A. Over. That's a plus one plus B. Very bad ones too. So we multiply this out and so equals X plus three. Uh A. S. Was to A. Okay, it was B. S. Must be Is equal to Ask Plus three. Yeah. Yeah. Alright, separated by terms around the S. Cosby S. So evil to earth around two. A. Plus would be yeah. Set equal to three. Yes. Yeah. Plus three. All right. So the vitality asses. I don't get that. Hey. Plus B. It is equal to one and then we'll bring that over here. She was counseling out of a plus B. Yeah. Should be cool. Which 1? So we're going to do is subtract this from the top because we only have one being each. We can cancel that for easily. So we're going to two A -1 as this is going to be a. And then B minus B is zero B. And then three miles one is a single too. So now we have that A is equal to two. So since we know that A plus musical to one, B has to be negative for me. So we have these values, you can put them back into uh decomposition over here. We have the two over it? S just one and we're gonna have a negative one. Yeah. Who were supposed to? Mm. Yeah. Mhm. All right. So um these two, the productions are pretty pretty easily recognizable as the applause transforms of the exponential functions. Right? Because you know that the applause transform of E. T. A. T. Is just equal. Yeah 21 over S my s a look. Mhm. So you know this. Uh huh. Two this will be times the boss transform of E. To A. T. And a. Here's our negative one. So it's the negative T. Um And I'll have minus and then it should be a little gloss transformed of each of the negative to T. Yeah. Yeah. So our answer is two ease and negative T minus. It's the negative too T.

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.

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.

So for white teeth is using the Cuban. Unclear why crime. Many times I just equal Thio Thio He mines t converting it. He's wife twice here. Three Raising the Austrians for failure as to why? Yeah, and why it will be equal. Three divided by three Teoh is one three as by Okay, Yes. Three. You too. Bye bye As three. What? One in light? Teasing reposting called Teoh T minus three t.


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