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Use the method of variation of parameters t0 find particular solution of the following differential equation_9y = 2...

Question

Use the method of variation of parameters t0 find particular solution of the following differential equation_9y = 2

Use the method of variation of parameters t0 find particular solution of the following differential equation_ 9y = 2



Answers

Use the variation-of-parameters method to find the general solution to the given differential equation. $$y^{\prime \prime}-y=2 \tanh x$$

Okay, so here we have a second degree, non homogeneous differential equation and our first of all righty auxiliary equation for the homogeneous part. So we can right here are square minus two are plus one is equal to zero. So we can right here ar minus one whole square. And that gives our is equal to one with multiplicity too. So here we have only one route, therefore are complemented Solution is see when it is two x plus c to multiply by ideas too X right. So here we have to multiply X right, Because here we haven't only one route with multiplicity too. Right now here why one is equal to eat us two x on why two is a call to X. It has two X now take the daily video off by one so derivative off others to access It is two X and now take the derivative off. X is two X, which is X is two X plus. It is two X right now find their own skin w which is given by why one into Why do this minus Why oneness into y two? So here are why one that is it is two X You're right To rest is X it is two X plus. It is two X Hear why oneness is it is two X And why to that is X. It has two X right now Buy Simplify. This we get it is to two weeks right now on the right set off the d. We have ah g o fax is a call to it is two x upon one plus x square. Right now our particular solution which is in the form off wipe, is equal toe. Why? To integration off by wanting to g o affects divided by W one skin minus. Why one integration off white to into g o affects divided by W into dfx Sorry DX now substitute the all value. So here. Why to is exit s two X Why one is Here's two x your geo affects that is it is two x upon one plus x square and here are W that is it has to two X now here. Why? When is it is two x y to his ex It is two x again here G off axis. It is two x upon one plus x square and again here Ws it has to do X now simplify this So here we have X s two x integration off You're it is two x multiplied by it is two x so that becomes it is 22 x and NT denominator we have also it has to two x so that will be cancel out. So here we have an only one plus x square And here also it is two x multiplied by it is two x studies It is 22 x and here it is 22 x will becomes allowed So here we haven't only X divided by one plus x square now integration off one divided by one plus experts So we can right here 10 in your sex Now multiply and divide by two So we can right here one by two Integration off two X divided by one plus expert into the X Okay, so here de levity off denominator that is equal to the numerator, right? So we can idea d by d x off one plus x square That is a call due to X right? So in desk in in this case, we can right here I learnt off denominator. Right? And here deal. It denominator is one plus x square. So that is Thea our particular solution now, right? The general solution which is given by why is it called the Y c plus y p. And here why? I see that it's your complimentary solution. Waas See, when it is two x plus, see to multiply by X into it is two X and they're our particular solution. Waas X It is two X into then in verse, a fax minus It is too extroverted by to multiply by Ln off one glass X square, that is theory. General solution. Thank you.

Played over todavia blinked. It's our problem of the 50 from the section it's actor review. Here we have a solid friendship equation. Why double less blessed to wired eyes plus lie? Because explain was you to the politics. It is D Squire plus Tour de Pleasant are recorded zero for us. We will find the temple for the child is a question this camp richness big plus learned that was quick So, uh, minus minus, therefore complimentary. First I will be Seaver here is to find a sex plus see through exteriors to my sex This can revival. This can't be right, right? No, you have to find w w because you to the park by the six eggs people about minus six Minor fetal apart minus six into the bar by an ethics My enough eggs eat about minus X But you will be getting us You know about my task to Rex? No, we have to find a Were they were because in the other minus, like f off. Thanks, but number you Phil minus x. You don't bother my no sex into Xing with people about its by you did a part minus two weeks ritually getting us my enough. You do that, Bubba looks the jury obligating us it'll about minus Parex by by the similarly they were equals intolerant by where everything's by W Why burn if people about my nothing why are fo fix? Exit was futile about X available minus two x But you're going to have in the middle of our directs by X the So the solution will be Seaver. It'll apart by an ethics plus he to eggs people about minus six. My left a war by you did about minus two x By group you go into the power bi sex Presby You are like I think they irritable Relax by X dx And do beware this exit to the part of my sex. That standoff. Our question. Thank you.

Hello to our trivia going Bizarre number number 51 from the section Doctor Review here we heard defying the situation for the differential equation while you never really. Why not? A two way plus by equals Get up our ex log x plus We were finding tender solution that is repress my double and should be squared minus jodi plus one equals So which can pronounce the U minus from the whole square? Recalled syrup. Because learned Kamala. Therefore my Z or peps equals sealer People about X plus seed to eggs fatal apart. It's why Bernie's each other products like always exceeded robotics. No, but method of aviation off Arabicas We need to find out for you. But if my bar why learn best like like cool Bess. If we saw the determined we were kept, it s people power to us. Nobody needs to find it won't ever equals in bigger minus like and for affects They were a part w this is Fo Thanks These he was up for you For minors eggs Beautiful X We endure you toe the products and let X they won't let you to the power through it into the bottom west Tear it apart. Works will get cancelled and we can try to dust. Buy less x and let Nick with June get it Does minus take the square by do a legs plus take a square by foot. Similarly beaver can be regardless immigrant. Why Apple affects the word by number You you to the party you to the products and that they were. But it is a part for weeks. Good best. And let X you got You're right. Uh, x x minus. Okay. Therefore, solution is it's even people about it, plus the brewing people about it. Yeah, but you won't wave with every square by four. You do the biotics minus X square by group, and it's you to the biotics. Plus they are. Why were plus Bieber labor that is eggs land eggs in tow. Eggs. You don't about eggs. Why X in new ex Delabar. Thanks. The restoration of thinking different application. Thank you.

Okay to start this problem first, we're going to find the auxiliary equation P of our is equal to So that's our square. Plus two are plus one single zero Diskin factor into our plus one squared. Just go to zero. So are is winning people to negative one with a multiplicity of to eight months of two. Next are complementary solution. Why Sea of X. This is gonna be able to see one e to the negative X and then plus C two x e to the negative x Our particular solution y p is gonna take the form you one Why one plus u two y two where Why one is going to be this e to the negative x and then why to is going to be the x e to the negative X you want and you two are going to satisfy the following equations will have why one u one plus why to you too? It's gonna be called zero and then Oh, sorry Prime. And then why one prime? You one prime Plus why? To prime you to Prime is gonna vehicle to and then we'll have our f of X, which is eats the negative X divided by square root of four minus x squared here. Okay, so then, ah, horny to Cramer's rule, you one prime is gonna be able to w one divided by W you to prime is an equal to W two over W where we're going to find w W one and w two here. So w is gonna be equal to the Ron skin. So that's equal to So we hav e to the negative X. That's why one then why one prime is negative ease of the negative X and then we have X e to the negative X and then we'll have one minus x he to the negative x as y tu prime. So here we get eats of the native to X times one minus X and then minus plus x e to the negative two x When we ah, most by this, this is gonna council out with this. So all relaxed with is e to the negative two X right. Next, we'll find w one w one. This is going to be able to It's gonna be this, except we won't we replace the first column by this here, so we'll have here eats of the negative X divided by square root of four minus X squared zero then x e to the negative X and then one minus x e to the negative X All right, so now this is gonna be able to will have negative X E to the negative two x divided by square root of four minus x squared. Then w two is going to be able to so again, the Ron skin. Except we replaced the second column with this again. So we'll have The first column is negative e to the native X than eat to the native X. And then here we have our e to the negative X divided by the square root of four minus x squared and then zero hips. So this is equal to this is just gonna be e to the negative two x over square root of four minus x squared. Sui, multiply this by this. And then this is just zero. Okay, now w one or sorry, You one prime. Now this is eagle too. So we had w one. Divide by W w one was equal to negative x he to the negative two x divided by the square root of four minus x squared. So then now Okay. You want Oh, sorry. And then divided by Eades of the negative two X. So then you one prime it is going to be able to negative X over square root of four minus x squared. So you want it's gonna be able to the integral of that. So negative X over square root of four, minus x squared dx According to an integral table, this is equal to square root of four minus x squared. We could also do this by trick substitution. But for now, this is just ah equal to co sign of or equal to this here. So then now you to prime. So that was going to be able to e to the two x or Sorry eats negative two x by a square root of four minus x squared and then divided by EADS of the negative two x so you to prime is gonna be equal to one over square root of four minus x squared. All right, let me make sure I have that. Correct. My looking of the previous oops page. Ok, good. So that is able to that. So there, now you to it's going to be the integral of that. So this integral Now, according to an integral table, this is gonna be arc sine so arc sign off X divided by two. So then now R Y p of X, this is gonna be able to So we start with you one times. Why once that's either the X times four minus X squared and then plus, and then, ah, our should be e to the negative X and then X eat of the negative x times you too So arc sine of X divided by two. So now our full solution Why of X is gonna be Start with a complimentary solution plus C two x either the negative X and then our particulate solution which is equal to e to the negative X square root of four minus X squared and then plus X e to the negative X arc sine of X over two


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