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Assignment Previov equation by variation of parameters. the dlfferential sin(x)...

Question

Assignment Previov equation by variation of parameters. the dlfferential sin(x)

Assignment Previov equation by variation of parameters. the dlfferential sin(x)



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Use the variation-of-parameters method to solve the given differential equation. $$y^{\prime \prime}+y=\frac{1}{\sin x}.$$

Okay, so it's off Would start by solving our home ingenious equation to a land a squared minus. I over C squared is also bounder. You could see plus dryness. I only two. So they get why Wages to see one sign of excellent you plus C two co sign of exhibits, you know, would solve our particular particular. We know that all of X is equal to to sign of X. So our particular why a p is gonna have the form. Hmm. Let's see a sign of X or two plus b co sign of X over too. Okay. Now, if you let this equal to what one political to white you determinative of why one crime is able to e Just think of one over to sign. Um, next light. You see, How do we stop this? You're gonna let this portion equal to my one. This portion here, you got to want to. Why one crime is you could too. Thinking of one of the two mikes. My two time is you could do one over to co sign of exactitude. So we want to find you private e and be prime a T. So that they got one over two. The crime of side that's over. Two plus one of its human co sign effectively to be fine Has a form you co sign of X Over too was be sign of X over too. You could to side of ex. Okay, we apply cream is rule. You get that you time is because you're too signed Squared of X over too. Okay. And we get that be prime musical too Sign of X. So we have our solution. You know that. Why did you go to sign of X minus? Thanks. Times co sign of X over too minus close. I know, Becks. Sign of X over too. Plus, he won co sign of X over too. I beseech you. Sign of exhibited

This'd equation using variation of parameters. So first we have to solve the complementary equation. I'm good. Not used to let her m m squared plus three m plus two cold zero. So even plus two plus one equals zero m equals negative one. Em equals negative too. So our complimentary solution c one e to the minus X plus C two e to the minus two x So, um am equals minus one and M equals minus two. Gave us the exponents on the E here. Okay, then in variation of parameters, what you do is you assume that the particular solution is you want times e to the minus X plus you to eat the minus two x where you want and you two are the functions that we're trying to find and to the minus X need to the minus two eggs are from the complimentary solution. Okay, so it's been worked out for you. So what you do now is you said you want prime eat the minus X plus you to prime eat the minus two X equal to zero. So you get to do that. And then the second equation is you want prime times the derivative of this, which is minus each the minus X plus You do prime times The derivative of this, which is minus two E to the minus two eggs, equals to this sign. Eat the X. So all the algebra has been done. If we would have taken the 1st and 2nd derivative and plugged into here and did all the simplification, this is the equation we would have gotten. But we don't need to do all that work every time conservative been done for us. Okay, so now I have a system of two equations that I'm going to solve. So I'm gonna rewrite them over here in meat form E to the minus X you one prime plus eats the minus two X YouTube prime equals zero, and the most second equation is minus E to the minus X. You want prime minus two e to the minus two x you to prime equals sign. Eat the X so you can see that if I add these two equations together, the U one prime will go away. I'll end up with minus E to the minus two x YouTube prime equals sign of each the X or YouTube Prime equals minus E to the positive two x sign E x. Yeah. Okay, so to find out what you two is, since we know you do prime, we have to integrate that. Okay, I'm gonna just leave the minus sign and I'm gonna integrate. Eat the two x sign, eat the X dx And what I'm gonna do is I'm gonna break up the e to the two X like this. It's the one x sign each of the X eat the x DX because I'm going to do integration by parts. And I want the sign to be Devi. So you is e to the x devious sign each the x times e to the x DX. Yeah. Okay, so then do you each dx, dx and then v the integral of the sign of you d you is the minus the coastline of U minus coastline. I'll be to the X. So this will give us minus u v or U V minus the inter parole v d u. So now we have minus e to the X cause I need to the x plus the integral of the coastline of you d you is the sign of you. All right, so you to it. So this gives us you two equals minus. Okay, because remember that minus sign right there minus each The x co sign each the X plus sign. Eat the X or eat to the X goes I need to the x minus sign Eat the X. They're put a box around that. So don't lose it. Oops. Okay, Now, if I go back to the system of equations right here if I multiply the first one by two and add them together now I get eat the minus x You one prime equal sign Eat the X Are you one prime equal sign Eat the x times e to the X So if I integrate that sign e to the x times e to the X dx Don't give me You won So it's signed you, d you So it's in a girl is minus the coastline of you So that's you too, or that's you on there. Oh, my gosh, I'm the worst. Okay, so then remember the solution Waas y p equals you want Eat the minus X plus YouTube. Eat the minus two x So you want we found to be minus the cosine each the x so minus each the minus X co sign eats the X plus you too. So eat the X cosine. Eat the X minus sign Eat the x times e to the minus two x so minus e to the minus x coast. I need to the X. Okay, When I multiply this in this, I get plus e to the minus. X goes I need to the x and the minus e to the minus two x sign each the x Okay, so those to cancel out So I get y p equals minus eats the minus two x sign Eat the X. Okay, So the whole solution then would be why equals c one e to the minus x plus C two e to the minus two x minus eats the minus two x Sign e to the X Let's check and make sure this y p works So the first derivative will be the first times the derivative of the second plus the second times the derivative of the first. So that simplifies in the minus e to the minus x co sign Eat the X plus two e to the minus two X sign Eat the X Yeah Y p double prime First derivative of the second, plus the second times the derivative of the first plus the first derivative of the second plus the second derivative of the first. All right, so now that and that cancel. So I get signed. Eat to the X minus What's plus each the minus x cosine each the x plus. Here's to to e to the minus x cause I need to the X minus four each the minus two x sign e to the X So then why P Double Prime plus three y p prime plus two y p equals sign Eat the X plus. Okay, I could add those two together three e to the minus X coastline e to the X minus four e to the minus two X sign. Eat the X plus three of these So plus three So there might minus three e to the minus x co sign E to the X plus six. Eat the minus two x sign E to the X. Okay, plus two of that equal Sign it to the X like he was supposed to. Yea, we did it. So the answer here

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.

Why Double crime? Why's he just thanks? Credible primaries. Why equal here? 1st 1 is equal of your r equals plus or minus one. So they're my age. The one e my eyes. You what, Dida. X Y p axes equal. Teoh wanna acts? Yeah. My ex was to look eggs e x d one prime of x means x two primary e back equals zero in even prime. A bag equal to men is be to prime in fact you cubic Thank you, Cummings even prying backs my ex was me to prime your next week. Okay, then my answer. And here you are able to get you're able to get to prime of X gold Teoh e of minus x 22 sine x and then be one prime affects the full two minus e of me sex hope he into the sign Like to view on extra bendy means e in a means eating x multiplied by sine cool sign by the two of active nd means e to the x multiplied by saying he too negative Extra bit of a four sine x plus co sign X and then inserted these in tow. What y p and Y p X would be you. Simplify to my sign x four directive and why acts would be the one D of rice exports. See, too the X minus x divided by


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