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I'll be glad if u give priority, I have 45 miny′′ − 4y′+ 4y= xe^2xSolve the following differential equation for initial conditionsð...

Question

I'll be glad if u give priority, I have 45 miny′′ − 4y′+ 4y= xe^2xSolve the following differential equation for initial conditions𝑦 (0) = 1 and 𝑦 ′ (0) = 0 using the method of undeterminedcoefficients.

I'll be glad if u give priority, I have 45 min y′′ − 4y′+ 4y= xe^2x Solve the following differential equation for initial conditions 𝑦 (0) = 1 and 𝑦 ′ (0) = 0 using the method of undetermined coefficients.



Answers

Solve the differential equation or initial-value problem using the method of undetermined coefficients.
$y^{\prime \prime}+y^{\prime}-2 y=x+\sin 2 x, \quad y(0)=1, \quad y^{\prime}(0)=0$

Okay, I have my double price. That's why Brian minus two. Why equal X sign IX. And they also have Why initial condition of zero one and why? No, I are cleared. Let's order lying. This to equal zero r minus R plus two R minus one equals zero R equals negative tube or a positive one. And the general form that looking at right now, it's why equals c one e to the negative to x c two Incan X So don't start with my, um Oh, no meal first. My thanks. So e have why a x plus be When I take the derivative of that I can't just a And when I take the derivative of that I get just zero putting that into my equation. I get it. Seagram the A minus two a X must be I can't a my new to a X minus to be e x, they're four negative to a X equals x negative too, eh? Equals one any he would having a then I have a minus two b equals zero and minus to be that would be a hero. I know a negative on half, minus to be. He has to eat. Negative one. Now, I'm gonna start with my So far, I know I have minus four and I'm gonna start with my Trig No, my Trig and might re junk region. Of course. White devil crime. Last wife. I do explosive. Why a good thing to wear Plus these signs next. I would Negative too, eh? Next. No, to the who's going two x and why? Doesn't look right. He was negative for a Yeah, my or sign loving that into my general. Right? Initially equation of there. I have the cutlery backwards. All right, so this has been going there, so I'm gonna have a negative for a minus for me. Sine x No action. That's minus to a a Trying to a huh? Less to be sign you X minus a two necks. Thank you. Or a sign. I next minus two a Oh, my Fine. Next, my two a who sign two eggs. This too fine, X. Now I'm gonna combine like my life terms. So I have a negative for a co signer to X and a negative to a co sign to x. So get negative. Six A. Who sign to X negative for be signed Needed to be assigned to a quiet, negative six bean sign two x minus to a sign to x to be those signs Two eggs equal inside two x already So then I'm looking at getting my signs. I want my signs together. So sign too ready next page I have. And negative too, eh? Minus six b equals one negative to a sign of two x minus six. Be sign of two X equal sign of two X and then I have negative 68 co sign of two x plus to be signed co signing two X equals zero negative. Six a. Plus two b equal zero. All right, so that I'm gonna multiply this equation. Three. I had a negative to a t one and negative 18 eh? 16. You won't see Rose, my tea's dropout and negative 20 eh? Equal one a equals negative one. Money is plug that back into the equation and I got be equal. Negative. 3 20 months. Ready set. Now my general equation is shaping up. Now I have Why a woman's C one Either the negative to x c two He mine 1/2 thanks minus 14 by 1/20. Oh, sign backs minus 3 20 Sign, Lex. And now we need to find our initial conditions. So we know our initial position at zero is zero is when x zero wise one. So one equals C one plus C two ended, and it ended it minus 14 minus 1 20 Now, I have take the derivative of the equation. So why crime? The wall? Negative two c one e Make it two eggs minus the two e to the X minus 1/2 minus 1 10 Sign of two X minus 3/10 Come. Something of two eggs and our condition and 00 for crime. So zero equals negative to see. One one two minus one, minus three tense. So my linear equations that septum of white Mormon I have 8/10 equals C one negative. Eight equals negative to someone seating and 26 wanting it equal. Seaborn, let's see. Two. Now I want to ask you are I want to add them. So I just switched the signs. So I am going to subtract. No, I'm not quite this equation. We get this grub. So this would be 16 when he hits common denominators and I'm gonna multiply it by making it one Make it is Thank you and adding them down. Seek, use brothel. I get three c y equals negative 16. Positive 26 gives me and over 20 multiply both sides on third picture to the three Hey, with C one equals 10 over 16 inches on six. When I plug one sick back in, I get 68 60th which equals two. 17 over 15 has reduced for action. Therefore, my equation now becomes one. My vision becomes why he won't 16 e to the negative to lex minus 17. 15 e to the X minus 1/2 X minus 14 minus 1 20 co signed to x by 3/20 CE Final two X

Okay, so here we have a non homogeneous differential equation and we have to solve this by undetermined coefficient matter. So first of all right, the auxiliary equation for the homogeneous system. So we can right here are square minus R is equal to zero now, factor out are so which gives r is equal to zero and r is equal to one. Therefore, our complemented solution is C one plus C two. It is two X right now on the right side of the d, we have X multiply by ideas to X. So let's assume that our particular solution is a plus. BX multiply by others to eggs. But as we can see here, it is to exist already a solution off the given differential equation. Right? So here we have to multiply X for the particulate solution. That's our particular solution. Is eggs Yes, two x plus B x square. It is two X right now. Take a dirty with you. Okay, now here we can simplify this. Why does he Why the situation? So here. Why? This is the colder It is two x plus x. It is two x multiplied by a plus. BX Sorry a plus to be not be X can now take a secondary radio. So here we can write y double dash that is equal to okay, severe we can simplify this equation. So he are coefficient off areas two x that our a And here we have also s o a place that is two way. So here begin right to a last You are coefficient off a district. That is to be also right. So here we can write to be can now substitute all this video in this different cell equation. So here, first awful. We have a wider bullish minus why this is a call to exit is to eggs. So your wider bless that is a call to the way plus to be multiply by eaters to x plus x years to eggs Monday by my A plus four weeks Bless, be expert evens two X And here widest that is equal Do negative eggs. It is two x last a Sorry here we have eggs no day. So your ex multiply by It s drugs. A plus to be plus p x square multiplied by it s to eggs. And on the right side we have makes it s two X right? Can I simplify this equation so we can get here? A plus to be multiplied by eaters to eggs plus x multiplied by eaters Two weeks multiplied by two B is equal to X it has two X right now Compare the coefficient off It is two X rights on the left side. Your coefficient off it is to exist a plus to be and on the right side your coefficient off it is two x zero And now compare the confession off X areas two x So on the left side we have to be And on the right side we have only one. So here to V is equal to one And by solving both this equation, we can get here. B is equal to one by two and is a call Do minus one Right now substitute both this constant in our particular solution. So our particular solution is why is it called toe one by two X square. It is two x minus X. It has two X now right through our general solution. So general solution is given by y c plus y P Now here are complemented solution. Waas, see one. Let's see too. It is two eggs and here But you know solution is one by two X square. It is two X minus X. It is two X Right now our initial condition was why off zero is equal to to. So substitute X is equal to zero and y is equal to two. So your C one plus C two is he called to therefore seven is equal toe to minus C two Right now take a derivative or forwards and those solutions. Right? So here you can write why this is a call to see to it s two x plus one by two X square It is two x plus x It is two x minus ex areas two x plus It is two X right and there were initial condition is why they saw zero. Is it called toe one which means substitute X is equal to zero and why this is a called one. So here one is equal to see to l a zero l a zero minus zero plus one, right? Therefore C two is equal to two now substitute the value off C two in this equation so here, too and too will begins allowed. So here seven is equal to zero. Now substitute the value off C one and C two in general. Solution. So our general solution is why is it called to to it is two X plus one by two experts. He has two x minus X. He has two X. That is to your final answer. Thank you.

Very first question. Seven. My double Brian minus. Why, Brian? Off my boy. Equal sign of X. They were given the initial condition when x zero wanting one and wife prime. Thank you, Nero. Why Prime equals one now I have bars. Where'd minus two are? What's five zero? Nobody's quadratic formula because I cannot factor. So I have the officer B two plus or minus this. We're route of Beast. Where I'm going to spare becomes four minus four times, one times Fine. No, over too. So I have to That's remaining the square root of negative 16 over too. To plus or minus I over to which comes one plus three minus two I. So I'm gonna start out with general condition. Um, being even the ex, I see one co sign No. Two X because that's my cooperation. I plus clean too. Neither the x sign. Um, two X Now I'm going to, um impose Why? Um, yes, a code sign. X plus Be a sign. Thanks. Why? Prime would eat wall negative a sign Be co sign in my devil crime equal negative. A co sign minus b. Stunning. And I'm gonna plug that into my original equation. So I'm gonna get negative, eh? Of sign? Thanks. No. Yep. Negative, eh? You my double prime minus B sign of eggs that I have minus two. White crime. So minus two negative. A sign X waas The co sign X was Bible. I was five. Mine's a thing. Next be sign. Thanks. Eat wounds. Sign of X. Yeah, No, I am going to get my signs together and my co signs together So my signs together would be I only distribute all of that first I have give a oh sign X minus B nine necks plus two A sign nix That's a plus minus to be Oh, time X. Let's five a Who sign eggs, That's why. Be nine x equal sign. Thanks. I'm gonna get on my signs and signs together. You do that? You're raking corner. So I have, um negative. B negative B sign. Thanks. Plus two a Hey, sign of x. Last five B. Sign of X. Negative one B side of X plus five. Signed Vick. I get to a sign of deaths. Thus, for B sign of X equals sign of X. Now I'm going to get my, uh co signs together. So I have negative, eh? Come sign of X minus to be from sign of X waas five A. Yeah, finer X equals zero. So combining my life terms I have negative to be the sign of X less for a I'm saying of X equals zero. So to a class for B s equal one be Plus for a that's equal one and negative to be plus for a negative to be all right Can a plus for B two A. That's for me I have to a let's work be equals one And then I have negative negative to be and would be Who are a yeah equals zero So I am one to multiply Multiply this by two My bees will drop out No more playing this by two Yeah, e a minus four b equals zero Re copying my top equation I have to, eh, That's four B equals one who I have 10 A equals a equal one. A equals won over 10. Now I put that back into the equation back in here, So I have for times 1/10 minus to be equals zero that this is war 10th for to Fitz subtracted from both sides. I have negative to be. He calls to Fitz. No negative. Multiplying both sides by negative one head be he mumbles 1/5. So now what? I have weigh him. Why? He won't see one. He won that e but it's really see c one e to the x co sign new X waas c two e to the X so I know to eggs less one hen hopes I necks noth one bit sign of X And now we have to solve for C one on they give us the initial condition. Um when X equals zero y equals one. So I hung in my zero I have seen one time e 20 Who's I kn? Zero. Did you hear WAAS see either the zero sign of zero That's 1 10 No sign of zero one nine of zero. And when I saw that, I get it. C one even This here, of course, is one co sign of zero is one. So I have seen one nuts. Um, either zero is one but sign of zero zero's. I don't have a seat to term. So this is zero um, 1 10 co sign of zero is 1 10 and want this time zero sign of 00 become zero. So I subtract one temp from one, and I get 9 10 equals C one. So that's been going for my C one now to find life, see to I have to take the derivative and substitute in my, uh, one for my derivative for my acts at one for my y and zero for my ex. So my derivative, my prime equals 9 10 after the product rule. So I get e to the X who signed you? X and I put zero in right away minus two. Sign to trend zero. Yes. Easy X Easy to the yeah sign. Uh, zero The seat, too. Times two co sighing of zero time two times zero times e to the zero. So my wife prime, of course, equals one one equals 9 10 times e to the x at Times co sign of zero gives the 9/10 minus two times zero plus one sign of zero around. No, C two times two Those eyes of yours one. So I have two times see, too. Oh, plus my 1 10 My stop. There I lost my 1 10 actually be a mine driven. Of course. Thing is a sign. The room I look, I really didn't sign. Look, the room. What? One there. I'm fine. Okay for this. So 1/5 time's co sign of zero, which is one fit. So now I'm going to subtract 9/10 and I get 1 10 equals two C two plus one. Then I'll make that one bed one. Um, there's a track that from moved sides. So 1/5 might want 10 minus 1/5 which is too tense. And I get a negative 1 10 equals two time. See to multiply both sides by 1/2 And I get sea to equal Negative won over 20 So that brings me to my final equation. But Wu, why of eggs equals 9 10 You with eggs co sign two eggs minus won over 20 e to the eggs. Sign of two x less. Want him coat Sign of X. That's 1/5 sine x Uh,

Okay, So if you separate the variables and integrate boats and you end up with the integral off Y plus one divided by why why is equal to the integral of X times the side of X t X. So, for the left hand side of the first thing we're gonna do is we're gonna separate this out. It's become integral of wine D y. Plus the interval one over wide light is equal to the integral extent side of X gx. Sophie left inside. That's why. Plus, Ellen of the absolute value of white is eaten too. And then for the right hand side, we can use interrogations by parts to end up with negative X times, the coastline of X plus the side of X plus c. So putting X is equal to zero into the original equation. And since we know that why zeroes equals one, we end up with water. Plus, Ellen open what is equal to see which results in cuz equal to one. So therefore, our final answer is live plus ln off the absolute value of why is equal to negative x Hammes, the coastline of X plus the sign of X sign of X plus one


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