5

82 It.e?(t) 0a dt0...

Question

82 It.e?(t) 0a dt0

8 2 It.e?(t) 0a dt 0



Answers

$$ 8 y^{\prime \prime}-2 y^{\prime}-y=0 $$

So if a question A I want to determine whether this equation has modernise our newly lier coefficients or of the form G of X plus B y. So to start this problem, I'm gonna go ahead and just see what I can write. This s o I can identify what type it is. So my first up here is I'm gonna go ahead and get see? Why do you say that? So do that. I am going to go ahead and subtract this over and and divide by De Seda. And so when I do that, you should end up with why do you say the sequel to Negative? Why Cubes minus state A Y squared over Tuesday, the squared y. So it's just from rearranging the equation. And then I see that I can probably get a homogeneous equation from this. And to do that, I'm gonna go ahead and multiply top and bottom of this by one overstate Acute. Try to get that. Why? Over data on everything. And so when I do that, I see if I distribute this that I get, why overstayed a cubes minus why over theta squared all over two times. Why overthe Ada. So this is of the form the wireless data is equal to of data. Why So therefore this is Marchioness. However, I can also we write this in a different way. So looking at my star here, my starting with star can be re written as like you'd minus beta pi squared. Plus Tuesday's word Why do you want these data equals zero and all I did there was divided each turn by De Seda And so sorry for D y de Seda. I get that do y d theta is equal Thio negative like youth plus data y squared all over Tuesday two squared y and I see that I can separate this into two different factions. And when I do that, I get that my first term is negative. Y squared over to status where my second term sand flies out to. Why over To say that. And so now I say this is gonna look like a Bernie Lee. So I have moving this over. I'm gonna have my p of X, and then this will be my end up being my cue of X. So rewriting this into the burn really form I get do you? Why's data minus this one over to state of times. Why is equal Thio left, which is negative one over to state a square times y squared. So again, this is in the born newly form This negative one over to data is my p a fajita. This negative one over to theta squared my cue of data and then my end is equal to two. So this is also a friend, Julie.

Science squad. Why say teda, ladies? Why would you bar way? Do I will own division Sinus body citi dot the every days integration. Why do people do indication? Sinus file teda cause Teoh GTO. So we have the sequence to why four minus y minus one minus integration one include bottle minus one or minus one de y. That is the addition. Science, far deed. Uh, called steed. Uh, do you do so dishonest Mia's You sign the d Umea cause Tita major dying Will d d Uh, it is minus y minus plus immigration Ubar minus y in the way it is given as a relation the Esquire meaty. Therefore, we have this equal to minus why my next week for my ass when it is given, as he did about the over three plus the therefore further, we have this integration people's too minus why you bar minus my ass. You for minus. Let's do signed a power three Tito, our three I love seeing. So this is the answer

We're gonna take a loveless transform on both sides to solve this different communication once we have why of ass and which ones from the bat to get wet. T So that's the equation being just one. Do you, Bess? And the distance boarded up. The transport of sweaty and this stance Ford love interest one of GFT. So from here we have a true vets equals two y of s over geo bats, which is one nowhere squirt waas a minus nine. So hence we have h of t equals to the immersed. Look, let's transform official vaz That would be look a subversive 1/6 sounds one nowhere. Ice minus three minus one of us plus three. We, um, be composed this part reflection. So his for this part is eat with three t and for this parish needs the next 30 coefficients 16 So anybody six. And this is, by definition, science 13 over six. This is we have this characteristic conclusion. Lost four minutes 90 So committee composed prefect arised as I'm Armanis threatens or close three. Well, Cyril Silver white Cape T General solution equals two C one into 30. Let's see to eat in the authority. Plug in the condition we have White Cape zero Liz too. And the working from of zero is zero. So we have C one and C two. They both equals the both equal to one. And so the last step is to establish this identity while t equals two by definition, age composing with G. See, that's why Katie So the answer being son sh t composed with gft waas No one, sir. Here plus two ton skills challenge Pretty. That's the answer.

So in this exercise we are going to solve the following a differential equation by using block plus transform, which will allow us to convert the derivatives to a multiplication. So by applying the LaPlace transform to both sides of the equation, we get this expression and then, using the notation on the right and the formula for the leopards, transform off a derivative, we can rewrite our equation into this form. Now you would like to regroup all the terms containing the capital while vest to express its value. Then, once we get the capital wire vest, we would like to apply the inverse LaPlace transform to find the solution off our original equation. So let's apply the inverse LaPlace transform, and we will separate the fraction into two terms because it will be useful for us to compute the inverse LaPlace transform off each term separately, and this possible because off the linearity. So let's focus first on the purple term. Actually, what we would like to do here is to get this fraction into a form that we can easily recognize. There's a lot less transform off a function, and we are going to use the method gold partial fraction expansion. So I hope you remember this. And thus we will obtain two fractions, halving only leaner denominators. And we will be able to use the formula in green. So let let us do it. It's, Ah, Onley Simple computation. We can easily find a value off the proficiency and be so be It's minus one and is too. So let us separate the fraction into two fractions. And again, I think, soon linearity. We can compute both terms separately and using the green formula, we find the inverse applies transforms easily but help of the rector. This is more complicated as it is a product of two functions that, luckily, we have something Ah, theorem that allows us to express the inverse LaPlace transform off product off the LA Plus transforms, and it is actually equal to the conversation off the original functions. But now we don't know what's the function F. So it does express it. The capital F off us can be also expressed as a sum or difference of two simple fractions and then, using the same formula for the l A process, transform off into the power off minus offorty. We can compute f. And now, using the definition off competition off two functions, we will find the expression off the red term. Finally, by summing up both breath and purple terms, we find solution off the original equation.


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