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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.$$y^{\prime \prime}-2 y^{\prime}+y=2 e^{x}, \quad y_{\mathrm{p}}=x^{2} e^{x}, \quad y(0)=1, y^{\prime}(0)=0$$

Calculus 3

Chapter 17

Second-Order Differential Equations

Section 2

Nonhomogeneous Linear Equations

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So you have the full in equations? Cygan, Order. Torrential equation Y doble trying Minor Still. Why time Plus why Cold too to did he x? Yeah, even. Ah, this particular solution Excess squared dx sitting on the solution. What satisfies there? Conditions where zero is equal to one. And, um, the chief, uh, zoo. So is that Michelle? So, um, so the more general solution, Giavarini, this particular solution plus wise reach my genius useful. So the dental genius, we mean better. This is This is linear. We can explain it. This is so poor that wise, which we have. Well, Brian, we just avoid brain. Plus why bad it was zero. So this is their Virginia system. All my genius so well, we I hope that the particular if we differentiate that boys, we truck Japan's they use us. He's Martian there. So will shape your diary strokes. You have this potion. Why crime? These are people Mention this depression first These terms So two x Is this chicken? I don't see the X. Well, now now keep this one constant differential that So we're told this is just three exit will be excess square GDX. So this is the first Everything. Well, secondary for the 70. Ready. We would have to differentiate these first. So they're doing taxes too. About them secretly X Now we give these on car site, two eggs. Differentiate these. Which is gonna be Just get So this stands for the little guy. This trump? No, with double vision, we different shaped. There's this first term. So that is wets job. Now we differentiate. This term, it would be one. Keep these one constant through the excess court. Yes, until all 75 things will be to the X plus. That was that or x Make the eggs last. Sex is for the X, so Well, it's plugging. That's replace here with these expressions. You get that? So yeah, Second reading who have least so you should be zero x plus four bricks The X los excess squirt the X minus two times why I'm so why prime disease. So the miners wires so x x lost sex is for you Get now. Was the function itself Why sippy wasit iss this You need blood That blast X squared dx So we'll see you find these these over here with the two I'm Suzy minus four eggs. The X is this down? This term would be minus three. It's squared V X. So me, we have these terms for exit DX. These councils with that term are you staying on bad one? We'll cancel each other on their web. Eastern, we're gonna have so x squared. So these company factors everything us uh, you're Yes. Oh, plus that sexy scored. So the rex better miners, these one so be are currently dx two B minus two x squared. And then this one also acid reacts. So plus x squared. I know that multiplied by gets on the wall. We cannot This year there is gonna be X squared minus taxes, squares. So that would mean those right now minus sexes squirt on thes minus x corps councils with that. So you from the PDC? Indeed. The right inside or here. You saw that dry inside toe. He's so the particle personally question. There are people now for down with genius. Uh, since we need to Well, the general solution, um, we can dio under general backing provision the corresponding full. You know, you're here for the general for patients. Uh, yeah v r R Square minus two are plus one. What are wrong? Our square minus who are was one. This is a very assuming that you form a solution for commission off these terms. So have these. And then this term can be but can be seen us are, uh, probably do on land is one minus minus 11 So, yes, he does so well, since he's one of us particles one of this illusion. So you know, I'm a genius. It's gonna be that will he has ordered to. This one is order took so there. So she totally homogeneous is gonna be We have a combination a the x of Venice's orderto. So plus, we'll goes on me. Well, be X. So this is the Virginias. And then well, we know that our our particular is X squared. So the general solutions RV exist squired the X was that? Plus, here we mix. Plus B x x, you know, um 0 10 years leasing initial data to see you. What are these constants? A. So why that's you is not being used by on the following that you, uh that zero plus a your house. In meantime, serious off course here, so Well, this has to be one. He asked people to want to have news that a single one. Now you have this function. Why burn through people? Well, Brooke, for rule again brushing his first extensive picks. No keeping the cost of X square from their GPS off this time. Oh, wait the same yet for dinner that there won't be any apply current rule again. So yes, it is X not differentiating so Plus so now we have always these wired primal zoo will be zero plus zero plus a glass being you do this year cereal last year. So it has to be people by this commission, so Well, that means that they will be my But these one means of my nose when b is minus one. So for these how about the solution? So best solution with the to 81. So yes, these minus one for minus next has that So plus X squared. So this is the solution Finding something's find these Alveda. We're here, baby. Two year old issue on actionable that 01

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