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Verify that the given function is a particular solution to the specified nonhomogeneous equation. Find the general solution and evaluate its arbitrary constants to find the unique solution satisfying the equation and the given initial conditions.$$\begin{array}{l}y^{\prime \prime}-2 y^{\prime}+y=x^{-1} e^{x}, x>0 \\y_{\mathrm{p}}=x e^{x} \ln x, \quad y(1)=e, y^{\prime}(1)=0\end{array}$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

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So you were there. Fulling, sickle order differential equation while right. Minus food. White prime. Plus why you go to you the X over X. You're getting the all these Lisa helps for X equals you because, as you can see here, um oh, yeah. You have only bay or solution to be. They find, uh, or this year's here for all the reels where zoo you want for all the reels, smaller as you were. That direction is because I have zero Suzanne. He's not the fine. So the situation doesn't well defined X equals zero. So make sense of it. We have to choose either positive side organ said would. Besides, we just decide toe. I will begin that we're giving up articular solution. Why should solution is this? Ah x g X times mantle room affects all you can see. Martha Loreal Banks is the fine or so. So we need that tradition on Devaux with Chip first of these wise K. Klytus, Five questions. So that means that if you differentiate place, I said being minus partners function blood the function minus truly relatives of the bunch of budget religion itself. Yeah. So this is why why sippy, prime zombie or even ship is one first. So that OPEC's is one would be one of them seem to be eggs and not below it Operable three things you have to do They a threefold purple. So amount me friendship the certainly the x so that the x times come see three X that is a veritable feast down with mix That was not formally no different shape the last time we're gonna get on one of Rex the last time x t x times one rex on these things, Basil, because this expression will since x x positive is well defined inneractive well defined or explosive. So, uh, boss of white Brian the sample, This is just FedEx Now, the seven dysfunction you mentioned that we get, Let's well, we differentiated not really get three x times one over its on. These two are there now for this term. Well, this is the function itself so that we will thing these same line these same three times for just get a people Peaster. So what did they know? This term, you'll have be saying expressionist there, so I'm just gonna be that So you three x it was not for a little bit. Last X gifts. What? Plus ex convicts off a little bits. Last navyvieques Well, now for their broke into the ex when you're the last time we get a new experience for you. So that way, please. These nations is one of the well, his first term be the ex Im smartphone with picks was us about German plus Next. Thanks. Um, plus DX not roll over Rex. It's not Lex. Um, plus six six, even sex. A lot of a little thanks. Plus, we'll Kweisi Weeks flies bx and then also simplify here. Ah, these terms on these other terms are the same. So it's just gonna be 2000 and just you raise that term. So this is, uh, this that is those Those four terms are this person now, uh, my friend to be minus twice the X and not for Biggs. Oh, plus x You? Yes. Last three X Now the function itself while these x to re plus x x x So we do it. The solution? Uh huh. Incomes these terms. My house is gonna be minus two times the same thing this time cancels would be held? No, this there all this term it should be. You should stay because this is their arrived outside of the question. So if you see, there is no time like these one. So that is good. No, this term you could be X. Oh, so these line will be well, see, so we can release conservation. These Utd X as a reminder. Stupid BX one so that we end up you know, Mr X, what was that term? Plus it's angry ex special bricks minus two times this term. My school was that excuse The next half of a very complex. So these three times cancel boys times a positive. But this was science about councils with that. So these indeed source the articular. Well, you can see we're now for them with genius. Uh, by the river. Up on the general coefficients. This is a question Will be revealed Are square in my school. Are That's what people is here for the then what genius? You make a solution on the form you do. They are dicks. So these leaks putting on all these are my rooms. Sorry it was one these that so better. All the solutions will be baby pics on since he's your vision with one double plus b cost lost you folks who looks one. So, uh, off the we need to satisfy our John data. Yours are some initial, very other satisfy. So these if the girls were dis Constance because the general solution with being piece your genius was the particle arbutus eggs. Next, my final. It is partially what genius or would use is the particle. So we're even that, um it satisfies the conditions. That why I wanted to hear the development fortune one because he's going to see. So you have dysfunction. Will you punch out one? It's gonna be Hmm? A she wants so a dance. Oh, I see. Plus b, there's one them. See? It was you on the inner. Plus, he that's not really one. I would not want this issue. So this part with me gun and so we'll have to ask the vehicle toe on them now for the people of the function, but one. So we're doing this conference people toe that is just a with the ec's down off these will be pressure is one first. So he comes into the x being irritable yet now they developed the second time off the people X with X Ramsey decks another their business, they just pop. It is, uh we get it here is gonna be in a db x not for love In all flicks Last year the x x plastics plus X You did the x not full of rhythm six x natural a little on the last term. Waas GDX Velocity X Um, so this all these We can start with this one on the gable with that one. He's got the, um, ready one. The the gloves A dams. Last dance you. Oh, damn. See? Plus B there, there won't matter a little more serious. So last hero plus natural ones. You plus years about two people. Because you so, uh Well, uh, good. Divide off these equations. Should I ve attended the equations A plus, The one the question would be to be They want us to do glass. One more on Daddy. Siegel is here. And so we were very close with this town. It was Teoh one. But we know that this term being one. So this is gonna be one. I'm there now. We would end up well. So here we would end up by doing that. Replacing again, Doctor One plus me. One issue for zero so well have bring plus two people so municipal toe minus toe on them from these, you know, about play people toe one minus so that they would be well to one minus my store on They are physical Do three. So the general solution was being, um it's got the three to be x ah b is equal to minus two. So miners to x 30 x x x nothing Plastics. GDX Marshall over X. So these should be the solution. You hear over what bird does constants? Hey, being under she's

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