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Using the method of Key Identities find a particular solution for the equation below:y" - 2y' +y = e'...

Question

Using the method of Key Identities find a particular solution for the equation below:y" - 2y' +y = e'

Using the method of Key Identities find a particular solution for the equation below: y" - 2y' +y = e'



Answers

Solve the differential equation using the method of variation of parameters.

$ y'' + 3y' + 2y = \sin(e^x) $

Question 15 we start with why Double brine minus three. Why, Brian? What's do why equals even the X. What sign of X? Because we are square minus three r equals zero are no minus Oh, are mine r equals two one which gives us my equal. See one. I mean to the two X love seat too e to the X and we try Why? Yeah, ones ang even the eggs. But this term and our standard being co sign x c sine x, our sign extra.

Question or our mission is 12 let's do lie. No. So I have are spared too. Plus two, he see grand actor that do formula with four. That too over a two which it's moving chain plus your which kids? Two. After that, he lives too. I too. So that gives me the form, huh? Why? He would see one, uh, e to the X. Because I haven't I my the ex Who's I nix plus e to the X C to sign And then we continue our work to find out We'll be adding that So I have, um, my general why it's going to be a X Hey X love b see e to the X by wide pearling. Well, then be just a with us c e to the X and my double crime is just Now what I plug that into my formula here have been started on a new change. I get it e to the X did my double problem minus two a class C to the c E to the X less too a X, but be thus c e to the eggs won't necks. Well, I need to the ex so I have seen eat to the ex minus two a minus two Seeing e to the X less to a left To be less to see equals X and then undo it. My aides first. So my aides I'm saying that my ex is first I have 282 a x equals X therefore, to a has to equal one and a equals 1/2. No, I'm gonna get my, uh these So I have my no exes. So I have negative Negative too, eh? Plus two b equals zero My A a negative 1/2 negative. Two times one hat plus two b one be. It was one. And then I have might see e see being so I, um see, I need to the ex equals e to the X Therefore see has to eat one. So now I have all my co affections and I can write out the general of my equation. Though I have my see, I even the experts I could put this in parentheses, but I you to the ex c one co sign X plus Evening X. See you signed X. Yes, I wouldn't have. Thanks. That's one head. Yes, to the ex. Because that was one That is my

I should buy our problems. Why? Devil Pride minus four. Why, that's why equals two negative bets. So this becomes R squared minus four corner. That's fine. Equals C room. It looks like you're factor, but because we have a plus five means signs of the same. So we'll factor. So we're using our quadratic formula. So in my quadratic formula, I get forward. Western Linus, this where 16 minus one over too. Which gives me home. But we're plus reminded this where you have negative for over two which that's three minus to I over too. Which gives me two. So because we have that I are to become the exponents or r e. So they read the first part of my equation will be Why eat waas e the two x c one sign X. That's the Oh, and now I have to I think I can write this all this is not that long. I have I want my it's going to eat wall. I'm gonna try a to the a e Negative X. Why prime then would get me negative, eh? E to the negative. And why double crime, of course. Well, then, give me a e negative X. I'm mending my equation. I get a e to the negative X for my double prime buying it for and negative, eh? E negative. What's the high? A e to the negative equals two negative effects. So I have a duty to negative. Next. We're a e? Yes. Why? A equal negative X. So I have 10 a e to the negative exes. Eat with E to the negative X. There were 10 a. That's the one and a equals 1 10 So now the general form of my equation actually go back here and just add on nuts one can you to the negative x.

Hello. So today we're gonna work on this problem. And so what it is is the second derivative of Y minus two Y, uh, minus two times the first derivative of y plus why is equal to e to the power of two X? So, as you can see, this equation has two parts to it. There's the left side. And then there's the right side. Now that the right side has just a zero like only a zero, this would be a homogeneous differential equation. But because there is, uh, something on the right side, it is not homogeneous in the solution. Will have two parts to it. So the solution since it has two parts, it'll look like this. Why of X is equal to why sea of X, which is the general part of the solution. Plus why P of X, which is a particular part of the solution. So the general part of the solution goes along with the left side of the equation the blue side and ah, particular side. The solution goes with the yellow side. Eso What we're gonna do is first, we're going to solve for the left side part of the solution to do this, we're going to get We're going to derive a complimentary equation to help us find a solution. And so, the way we're going to do this, we're gonna use the variable r, and then we're going to copy the coefficients of the blue side of the problem. So if this first coefficient is one right, then it's going to be one times r squared. And so, as you see, we're gonna use the variable art, and it's gonna go down by degrees. Okay, so next we have minus two. That's a coefficient, right? So minus two. And then this is only our And then here we have a coefficient of one. So will be plus one positive one. We're gonna set this equal to zero, and then we're gonna solve for R and find the root, which in this case, the way we're gonna do this, um, are is going to be equal to one. So there's only one root for this general side of the equation, the blue side. And when there's only one route Ah, there's a fear. Um, that states that when there's only one route the general part of the solution well, look like this is gonna be see one times e to the power of our X. That's an art. Plus the two times x times e to the power of our So that's an art. Now, if there were two routes, uh, it would be R one and r two been this case r equals one. So actually, one times X, it's just x. And so one time sex is just sex. So this will actually be the blue part. Ah, of the solution. So that is this part, right? Nice. So now we can move on to the yellow part of the solution. The particular part of the solution. So we're gonna do this is we're gonna say y p of X is equal to see a times E to the power of K X. So if we go up here, we can see we can copy, uh, C equals one, right? And then K equals two, right, because she is just one and then or C is just one listen more like this, and then K equals two. So why P of X is actually equal to just a times E to the power of two x. Okay, so we need to actually solve for a and to do that, we're gonna find the derivatives of Wipe. So the first derivative of I P is actually going to be too times a times E to the power of two X and the derivative of the second derivative of wipe is actually gonna be four times a times e to the power of two X. So here we have three different things, right. We have YPF X. We have the first derivative of Y p of X, and then we have the second hoops. We're gonna have the second derivative of Y P of X. So there are three things that we got there and what we're going to do with this information is actually going to copy the original format of a problem. And we're gonna substitute in why and the first derivative of y and the second derivative of y. So what that's gonna look like we're just gonna have the second derivative of why which is for a times E to the power of two X and then we're gonna follow this right, we're gonna follow it. So it's negative two times uh, the first derivative which is to a times e to the power of two X And then we're gonna add just why, Right? Uh, we're gonna add y, which is a times E to the power of two X and we're gonna set that all equal to e to the power of two X and we're gonna solve for a And as you can tell, it's for a time. It's a part of two x minus and then this is going to turn into four a times e to the power of two X So it's gonna be four minus four. Ah, so actually these to cancel out and we're left with only this So a is equal to one and we're gonna substitute that in here. So why P of X, right? Sorry, Y p of X is actually equal to since a is one is just equal to e to the power of two X. So this is actually the second part of the equation. And to get the entire solution, wait to get the entire solution. All of this we have to add this and this together. And so the solution will be why of X is equal Thio. See one times e to the X. We'll see two times x times e to the X plus e to the power of two X and that right there will be your solution.


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