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(15 pts. ): Solvell ly = e*via the method of variation of parametors. That is find y = Yc + Yp"...

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(15 pts. ): Solvell ly = e*via the method of variation of parametors. That is find y = Yc + Yp"

(15 pts. ): Solve ll ly = e* via the method of variation of parametors. That is find y = Yc + Yp"



Answers

$23-28$ Solve the differential equation using the method of variation of parameters.
$$y^{\prime \prime}-3 y^{\prime}+2 y=\frac{1}{1+e^{-x}}$$

Mhm. Give him uh So we are rewriting our differential equation in the form of art squared -2 are Plus one. And this is equal to My next one squared equal to zero. So then it implies that Our it's equal to one. So White Sea of Eggs is going to be cool to see one. He exponent X plus C. Two. E. Exponent X. Experience E. So I have C. Two eggs E. Exponents X. So that's why one it's equal to E. Exponent X. Then why two will be equal to X. E. Exponent X. So then it implies that so it's in place that you have way one way to prime minus Y. Two Y. One time. This will be equal to E. Exponent X. You have X plus one. The exponent X minutes X. E. Exponents X. E. Experience eggs. And this is equal to E. Explaining to eggs. So then this implies that you one prime is going to be x. e. exponent X times E. Exponents X divided by one plus X. Sue the power soon X squared oh divided by he explanations to eggs. And this will give us mine it's X Divided by one plots. Eggs quake. So there are you one you win will be equal to the NC girl of this eggs divided by one plus X squared the ace. And this is going to be called to my next one divided by two. You have lain of one plus eggs suede. Then you too crime. So this implies that our youtube prime will be equal to he exponent X finds he experienced X divided by one plus x squared oh divided by E. Experience to X. And this will be equal to one divided by one plus X squared. So then it implies that are you too will be the inside girl of one divided by one plus X squared the eggs. And this is equal to son invest of eggs. So hank's a particular solution. Y. P. Will be equal to my next one divided by two E. Exponent explain one plus x squared less. Bless X. E. Exponents eggs so invest of eggs. Then a general solution. A general solution, you know solution for a differential equation. Y. X. Will be equal to. It's the power X. Time. See one let's see two X -1/2 lane one plots X squared plus X eggs. Sign invest of eggs as our final answer.

Today we have this problem which we're going to solve using the method of variation of parameters. This is a second our linear No, no Imogen 70 we will denote the right now. Zero rare inside by F X And as usual, the first thing we need to do is to solve thie How much years in question, which is simply the same in creation, with the right hand side being simply zero and something. This is very simple. We simply write down the character see equation that isthe r squared plus three are prostituting Ciro. And once you solve this, we will have and two solutions I want to say the one and art to its new two, which means that our two annual independent solutions for damages parts are why one is able to Internet e X, and why two is Turner to X. So we have This means we have these general solution for the generous part which would threaten us. Why want see one white one must see to why two So the only thing that's missing now is thie is way. We have to find a particular solution ritual call by wise api and they and the method of fishing parameters tells us that we need to look for this guy and turns off a round in the condition ofthe into solution we've just obtained. So we're looking for two functions experience such then this. Ah, this baby thing is a solution off this equation so we could go through all the algebra off by taking on the drilling through spread. Fortunately for us, there's a very well known formula for each of this expects. So we we can simply use that Oh ah talked in our solution. So they form a line I will, which I will write down here. So a off X is simply equal to negative and tiger to off Why two times f f x, the ex and derided by what we call the bone scan off our two functions. They reverted found so and so Let's continue this actually, so you know, in order for this, we need to find the Royce can. So, as you might recall from the previous chapter y one, if you have two functions, they run a scan is equal to its two by two insurance. Why want prime word to prime Andi that's convenient. This on the next page. So this is equal to remember why one y two were into a negative axe on DH to marry to X. So they're doing. There's our native E to the native Axe and daters Later, two times easy to XO computer this to writer determined. You'll see that this is equal to native each Max. So what we have to do now is simply Valerie dissented during. So now we know all the ingredients y two ever x, which is this? And Delhi. So it's student here. So ate off Max. Cool. Two negative and center itself. So I have y two times f here, so there will be too native to x times Signe Well, you need to wax the Ryan. We're in the wrong skin, which is just this guy here. So it's again negative. The 23 acts Yes, here. So after the nature science cancel and some of the expansion terms can so we see that this is equal to and danger into off e two ex Ein teo angsty axe. Now this. Now, this is a simple cal calculus to integral and their several ways to do this But you can probably. Or to see that if we use the u substitution with around with this. But this is a different Grable tear. So if assassinated the expression lexical to tea, then receive it. The tea is equal to the axe. Bruce, Charlie yaks the X so that this is a simply equal to sign. Sign it, t two. This is because our DT contains this eatery Exterminate! Sign of X. Simple becomes sanity. So now this is very simple. It's simply Nate. Of course I am too right away. So you don't have to put plus C here because since we're so if you put plus some see here than in the end, we write down the answer. We have to multiply this by why one So. So do you concede that this term's constant times? Wyman term is already, you know, taking care of and the have a homogeneous solution. So that's why we don't have to deal with plus c constant here. So that's just something to remember. And so let's move on to the calculation of BX. So it's just that it's sir a new bitch here. Yeah, so b X, it's very spit it taking away the same chocolate. So this is able to refuel? No. Well, and I guess I haven't written out from was its recently consumed you simply If this native sign and this will be why want so e x would be and endure it off? Why? One times f of acts times uh, the dumping. So why one was Inter native Axe, This is Signe off ee to axe DX, derided by nature's being there three x so after cancellation, this becomes negative. Two to x Sign of X sanity to ex Deeks. Now this already. Look, this is already looking very similar to you. Just that way won't simply that Substitute TV because eat axe so that we have did too sick or too e t x t x and so it will have negative. We'll have negative. So you see that we have each other to X here. So one So this will become t fire sign team T t. Because one one'd rex comes from DT part and one eater X from t part. So they so we'll have our three Turks. So now for here this again is a routine exercise for the integration private by parts method. So if we didn't have this extra t here than we could directly involved at this. So our goal is to get rid of this team by taking its through into which is one. So suggests that we think we do the following intuition re parts. Um, you is in court to Teo e sitting cool too. Sign off, too. Do too, right? Actually, it's put in their sign here. Jasper Hammond, later scientist. So then this would mean the U. S. Simply Teo. They were due to have tea is just once. So take a look at this one here and so d From this, we can see the V is simply co sign off team because director of co sign is nature sign. So no, no. So now we'll have This is simply you times fee, which is two times coursing. Tio the time score sign to you. Minus three times you sort This would be integration course. Sign off too. Okay, team. And that this is very simple. It's just entering. Dissented, and it is just a sign of so some rest that here now have tea co sign two minor sign two and remember to see their acts. So we had to down invert over back to our x variable. So, you know, I have a sign off the ax or dismember. This is R E X so we can write down our oh social y p now. So this is number. So this is X times y one plus beer sounds right here. So why one and way too, Zaheer and a access their co sign t y gets different here too. That right? Everything in terms ofthe x variable. So this is my X. So why, Why he would be Why? So it's just right down here. X y one less yaks. Why two on X is Native Curse sign being axe tires. He terminated axe and be Xs here and multiply that by right eternity to excellence Prentice's Yeah, thanks. Cor sign. Really secure co sign? Yeah, thanks Minus sign off. The X times are y two witches Need two tubes now. Interested? There's usually calculation. So it's It's recommended that when of whatever you've got here, you should open the bracket sent and look for cancer. And indeed, in our case, we have one because we have three X here and Internet to exterior ritual becoming through Native X Times, Carson E X, which will exactly cast with. So I'm going to use a different card to emphasize the cancellation. So this term will cancel with this and this which will leave us with only one term. That is negative E too. Two X tires sign off ee to ACS. So this is my particular station wide piece. So if I write down two final answer, you will be simply Why single too? I see. That's why particular c one e during native X plus c too inter tuner to x minus. Um, well, class white people in this was minus dysfunction Certs minus E to native to x times. Sign off the X and this will be the final Astor for crawl.

Using the method of variation of parameters We have Y. one. So I want to be equal to e. Exponent minus X. Then why to to be equal to e exponent -2 x. So this implies that why one? Why to prime my next Y. Two Y one prime will be equal to minus E. Exponents three minus three eggs. So then this implies that are you one fine? You want crime. It's equal to mine. It's you have signed E. Experience eggs E Experience -2 X. Divided by minus E. Exponents minus three eggs. And this is equal to E. X. Sign mm eggs. So it implies that. So yes this implies that are you one will be the integral of this? So you have the integral of E. Exponents X. Sign eggs the eggs and this will be equal to minus costs into the power X. Then our Youtube prime. So you to crime. It's also equal to sign E. To the power X. It's the power my next X divided by minus E minus reach. E exponent three X. And this is equal to minus is to the part two. X. Sign E. To the power eggs. So to find you too. Youtube. You too is going to be the inti girl of sign. Each. The power X. You have eaten the part my next ex. So it's signed two will be the integra. Thank you girls minus E. To the power to X. Ein eat the power eggs the eggs. And this is equal to eat the power X. Cause it's the power eggs mine. It's sign it's the power Thanks. So then this implies that particular solution will be equal to minus E. To the far my next ex. Because it's the power X minus E. To the power minus two X. You have sign U. To the power X. Mine. It's is the power X. Cause into the power X. So then this implies that our general solution a general solution to a differential equation Y. X. It's equal to you have see one minus cause it's the power X. Times you to depart my next eggs. Bloods you have C. Two minus sign each. The power X. Plus eat the power eggs. Because because E. To the power X. All times E. To the power minus two eggs as a final answer to the differential equation.

Even you have what I want. I want to be equal to Each the power -2 x. Why to its equal to X. It's the power minus two eggs. So then this implies that's why one Why to crime mine? It's why do I want prime would be equal to? It's the power minus four eggs. Something Y. You want crime? It's equal to minus E. Experience minus two X. You have eggs. He exponent -2 x. Divided by X. to the Power three. He explain it's minus for thanks. And this is equal to my next one divided by X squared. So then it implies that are you one is going to be the insignia of when divided by X. Squared the eggs. And this is equal to X. To the power my next one or 1 divided by X. And you too. You're super prime. It's equal soup. It's the Power -2 x. You have each the par -2 eggs Divided by X. to the Power three. Used to the par -4 eggs And this is equal to one divided by x cubed. So then it implies that you two will be the insider of one divided by sq the eggs. And this is equal to -1, divided by two X. Quaid. So dan a particular solution Y P X. Is going to be equal to eat the power minus two eggs divided by eggs. Mine. It's eggs E to the car minus two X, divided by two eggs squared. And this is equal to It's the Power -2 x. Invited by two X. Then a general solution would then be so I Jenna a general solution Jenna solution Y. X. It's equal to it's the power minus two X. Sometimes you have c. one Plus c. two x plus one Divided by two x. As our final answer.


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