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Consider the nonhomogeneous DE oyl (c + l)y' + y = 2x2er and let %1 (a) er be a solution of the homogeneous differential equation;(a) [4 Marks] Determine a gen...

Question

Consider the nonhomogeneous DE oyl (c + l)y' + y = 2x2er and let %1 (a) er be a solution of the homogeneous differential equation;(a) [4 Marks] Determine a general solution of the homogeneous differential equation:(b) [8 Marks] Solve the nonhomogeneous DE

Consider the nonhomogeneous DE oyl (c + l)y' + y = 2x2er and let %1 (a) er be a solution of the homogeneous differential equation; (a) [4 Marks] Determine a general solution of the homogeneous differential equation: (b) [8 Marks] Solve the nonhomogeneous DE



Answers

Solve the given nonhomogeneous differential equation by using (a) the method of undetermined coefficients, and (b) the variation-of-parameters method. $$y^{\prime \prime}-y=4 e^{x}$.$

Hello. Today we're going to solve our problem Number 27 from section of you here. We had to find solution all the different elocution by using method of Yiddish off perimeters given because you have my number minor full. Why? Because five u to the power X. So when you find the general solution, that is by double Eskander, Liberty Square. My enough full. It goes so difficult. April's fool B equals Russell minus two. So dinner for the variety affects because Seaver, you know about off the books, plus single it about off minus two x here. Why you want? Because you feel about off works rightto because it is about off minus two x for you. The blue can be found out by W because you know about off the licks. This difference initiation. You prove it this to collect eres tu minus two x minus. Do it over off minus two weeks. So it comes to the line enough. Think so? Similarly, we can find young girls in the buying a slimer. My enough sort of minus light. Bush comfortable minus two legs in two f offense. Half off exes. Fiery Bella politics by, You know about all thinks they were by W W is full minus full for a trick on meth. Find by four In general, it apart off Minor thinks for it will be gone. Minus five by full. You know about my ethics. Soon it'll be in bed. Why run out for fixed by W Why one is you do about the works and for offenses fire you toe the products developed by the blues minus four. So it becomes minus five by four in the girl it about off three x It becomes my last five by the way. Into the bar 30 x So if I want to be right toe Yeah, I would be right to gamble it on us minus five by four. You know the boat off my an ethics into a beautiful but off lex minus five by Well, it is about off three X indeed. A lot off minus two. It if these are very good. My enough. Fired by three that it is particle a pea by pea off. Ex con trick laceration minus 5.3 year fixed. Therefore, torture solution is it's where plus disliked So just a plus minus five by three. And don't you so well a solution? What given differential equation, But you think matter off. Thank you

Okay. Good day, ladies and gentlemen, Um, today we're interested in problem number 45 from section 4.7. And what it's asking us is to find a particular solution to be given differential equation here. Um, given that little effort, T here is a solution to their on corresponding homogeneous equation. And admittedly, I'm going to cheat a little bit here. Um, s O I actually solved the first part of this in my previous video, which is bob number 4.7 43. Um, and I showed that if why one of t is equally e of t. They're why two of tea is just tea plus one. So if you want to know how it actually go about doing that, please just go and, um, wash that video. I did it there. So, um, now, with that in mind, now what we have to do is we're going to do is then used the variation of parameters message to solve this guy. And as I always do, I started used the same set up every time with the variation of parameters message. Um, the first step I do is I go through the various equations So first off the wrong skin, of course. And that just ends up being t negative. Tedy, the tea. Ah, And then I started right down my triplicate of equations here. Um, And from there, of course, now GFT is t each of the two t. Remember, when you get the g of tea here, you have to, um, divide true by the tea here on the why? So you actually end up with not t squared unity? You just have t e the tia's I have here in my GFT. So then when I set up the variation of parameters equations, um, you know, the I just get view one prime to be this and V two prime to be that. So now, once I have the one prime, the two prime Well, then it's in this case particularly. It's simple enough, but I just integrate back and I get view one of t to be here. Books. Um, sorry. View one of t to be this guy and the two of tea to be that guy, um, substituting back to get Why p And of course, we know already know why P is justice. And when you simplify it down. You're just going to get something like this, which is for C two e ah. To the two. T divided by two times one minus t. Sorry, that's the one. Clear. Um, but then we plug this in to get the general solution. We know their wife. T is just why people's y h. Um, this is why I p here. We already calculated that above. And ah, this is our, um uh Why t Yeah, yeah, yeah. Why? Sorry. So that's why h ah. So in the end, um, you know that that's our Ah, that's the answer. Um, it's Yeah, it ends up being simple enough, but it's still a number of steps. And I hope you don't think it's cheating of me to just use my previous, uh, work because it's, you know, it's Tze something I've already done. And it's just the previous you'll if you want. If you want to see this, please just go watch the video. It's, um it's right there. It's just a previous one. But, um, it's it's not really worth me redoing it, but anyhow, uh, thank you very much. Um, appreciate it. Have a good day

Okay. So for this problem we are asked to find the original differential equation and we're going to go ahead and start off by finding the R values. See our values in this case for the first time right here, we got zero for the second term, since there's an X value right here, we can conclude that it is in fact also zero. So what we have to say is zero is repeating, Okay. And then finally the last term, which would be this right here we have eight. So our values are going to be our are And AR -8 and this equals to zero. We can expand this out R squared times ar minus eight, which equals to zero or r cubed minus eight R squared and that equals to zero. And we can go ahead and replace these are Q and R squared with Y values instead. So this will translate to why triple prime and R squared will translate to the Y double prime And it equals to zero. And so this is the original differential equation and our answer

Now in this case we're going to take a look at uh let's see here looking at a bunch of differential equations and I'm going to do all these next few problems together because they're all very similar. So we're given some difference or equations and I suppose I should have written them down for each problem. So let me actually, so 36 we have S double prime plus four S crime. Let's see. S +40 Um Yeah and again this is notation is getting bad because again engineering this would be C. And this would be K. Um So but they B. And C here for generic differential equation. So we'll keep with their notation. So basically they ask for what find their values to see that make the general solution over damp. Under damped and critical. Damn well the what you need to look at is the discriminate except to discriminate anyway. Yeah, I think so. The thing under the square root in the quadratic formula. So b squared minus four. See these all have a in you know in the quadrant in the characteristic equation of one. So A. Here is just one so B squared minus four C. In this case we have B. Is four, uhh C. Is C. And so we need we have 16 minus four C. Is that greater than equal to or less than zero? So if it's if it's um over damped it's going to be greater than zero. Well let's start over damped. It is going to be a greater than zero. Why did I? Oh I think I looked at them. These are probably all messed up but I thought they said over damped under damped first. So if it's less than zero it is under damp. So that should be that. No I had it ready I think. Yeah. So I see going down here it's definitely not right because I know this is uh she's anyway I think I had it right the first place. So if this discriminate is positive meaning that that means we'll have real roots of our characters equation. Which means that well if the system if it's a mechanical system we're going to assume that it's it's gonna be under damned and so we'll see that the roots all gonna be um all gonna be negative. Yeah so uh let's see here just cause you're gonna have a negative B. Plus or minus. You know something that is going to be less than B. So we have um this thing if it needs to be over damped we need right let me just do it on a fight. If it's over damped we need this to be real numbers but there's to be positive. So characteristic routes to be positive. So this needs to be real numbers. So this thing needs to be positive. So that means C. Needs to be less than four now for it to be under damp this needs to be negative because that you get complex roots of our character situation. So for that to be negative C needs to be greater than four. Yeah. Yeah. Yeah. I'm getting confused because I'm thinking C is the damping coefficient? And I'm thinking well if the damping coefficient goes up that should give more damping and more over quick. But see is our stiffness can see as our stiffness because okay as the stiffness goes up. Yeah then that makes sense. And then for critically damped we just know that this thing needs to be zero so that means C. Needs to be equal to for okay there we go. Um I taught vibrations for 20 years so when I see, see I just immediately think damping coefficient. So that's kind of why the kind of for the confusion here. I got to remember that. That's not what they're using here. So in 37 we have s double prime, it sounds too square to S prime plus C. S equals zero. Ok, so and again I don't remember seeing. Is this difference? No let's see here, discriminate is going to be B squared which is eight minus four C. And we need to find out whether that's greater than zero equal to zero, less than zero. So to be to be um over damped this thing needs to be less than zero or greater than zero. Sorry? So that means he has to be less than two. All right now to be over there to be under damped. This thing needs to be negative. So we have complex roots of our characteristic equation. That means he needs to be greater than two and for critically that that's just in between these two. So this is zero. And so that means he he calls to now for the next 1 38 is let's see here S double prime plus six S. Prime plus C. S. Equals zero. I'm going through the same thing to discriminate is 36 minus four C. Now whether that's positive negative or zero tells us whether it's over damned under damped, critically damped. Over damped means that this is positive, which means that see needs to be less than nine for over for under damp that needs to be negative. So we have complex roots of our characters equation. So that means he needs to be greater than nine and for critically damped it needs to be zero. So she needs to be nine then 39. We have let's see here. S double prime plus B. S. Prime plus five S. And so let's see here discriminate is b squared minus 20 is greater than equal to or less than zero. So the first thing they are over damped. This thing needs to be less than zero. Oh sorry, greater than zero. So we need to be to be greater than the square root of and I'm assuming be as positive. So you don't really ever have in mechanical systems and negative Supreme constant. Although theoretically you can construct something like that but not in a simple way. Um So I'm gonna assume be as positive. So B is greater than it needs to be greater than five times the square to to you could also saying it has to be less than minus five times square or two. But again that's not really. If we're talking mechanical systems which would usually are when we're talking over damped or under damp then B is going to be positive the spring constant. Now if further be, let's see under damped B needs to be less than a square of five times square to to and to be critically damped. He needs to be equal to five times the square to to and now in 40 we have, let's see here, 40 we had S double prime plus B. S. Crime minus 16 S equals zero. Well now all of a sudden we have this negative spring constant. So what do we have here? Um We have B squared minus four C. Is b squared plus 64. Now the problem is, is I'm not sure exactly how they're defining over damped, under damped, critically damped. If you say it's over damped damped usually means exponentially decaying, but because the stiffness is negative, these are all gonna be exponentially growing functions. So I would say that you can't find a damping coefficient that would make this critic over damped because over damp means exponentially decaying. Um Well exponentially growing means unstable, so I would say none. Um The other answer is possibly all if they're saying any exponentially type behavior is over damped, but again, that would be really poor definition of over damned. Now they asked for under damped, well you're never gonna get this is never going to be less than zero, so you're never gonna get complex roots of our character situation here. So um basically there's no values of B that we could make this um critically under damp and likewise we can't, there's no values, p no real values will be anyway, that makes this thing zero. So again, we had to have no values of the damping coefficient that could make this thing critically damped. So I'd say the answer to all of these is none.


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