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Question

Let {6, *)Jwrp erder 4a} elemeat _ otMt idemity eleriest Juskly Anceo_ puibe budlerr OCo) + My? 4' Wh Ota):( . Sul Ingt (6,#) isdnerphiclet 4 € 6SfRse Contain (ri , +)elenentSipse Shut thatGaans eleneat BMarphicWth Ofa):l, tut Ofats dhedlal Jroupdemet wth Ota)=s .Shww Hat even grvr ( 6, #) cderinn"plfzeilker (bw, +) (ps,"

Let {6, *) Jwrp erder 4a} elemeat _ ot Mt idemity eleriest Juskly Anceo_ puibe budlerr OCo) + My? 4' Wh Ota):( . Sul Ingt (6,#) isdnerphic let 4 € 6 SfRse Contain (ri , +) elenent Sipse Shut that Gaans eleneat BMarphic Wth Ofa):l, tut Ofats dhedlal Jroup demet wth Ota)=s . Shww Hat even grvr ( 6, #) cder inn"plfz eilker (bw, +) (ps,"



Answers

$$y^{\prime \prime}-5 y^{\prime}+6 y=-6 t e^{2 t}$$

Okay. Good day. Ladies and gentlemen, today we're looking at problem number 11 here. And the question is whether or not we can apply the domesticated of undetermined coefficients Thio this, uh, ordinary different show equation here. And so I'm not really gonna, um, go through all the different parts of the undetermined coefficients cause I think there's probably 5 to 6 distinct cases. But it is important, I think, for you two know each of those cases because they tell you how to go about solving, um, the, uh, ordinary differential equations. They'll tell you how to solve a bunch of cases of, or a bunch of, um, ordinary difference, your equations on particular ones with constant coefficients here. And so the first thing is to realize that, um oops, sorry about that. So if I take three different, distinct functions, I think one of the year of two of two here at three of tea Now I'm going to really look at this case by case in each case. So in the first case, this one here is, um and it is in fact, a proper form. It is one of the cases, and I'm not sure which But if you look, if you flip through, you'll see the thing cases. And this is in fact, one of the cases, um, in the 2nd 1 here again is also a case again. I don't know exactly what's important, but it is, in fact, one of the cases covered by the, um, undetermined coefficients. But the 3rd 1 is not and in particular one over tea is not a polynomial. It is Tito the negative first, and that is not, um, covered by any of the cases. So in particular than, um, since you have one I mean, really, you can't get rid of this one over t s o. The end of the final answer is that it's not applicable, and it's not applicable because there's no case that covers us. And the older way you could solve this. Using the undetermined coefficients is to break this into threes. In cases here, I'm solve each one and then applies the the superposition principle. And in this case, you can't because one of those, uh, you know, is not solvable using that method. Uh huh. But it doesn't mean that there's not other methods to solve it. It's just not solvable using this method really, all the collections after. So there's no need to go any further with. So, um, again, I would just mention that, um, it's a good idea to have in the back your mind. What thes, um the what the undetermined coefficients method involves and sort of go through each of the steps because it's it's actually fairly involved. There's quite a few different steps, and there's a bunch of different cases. So there kind of TVs, but still probably could know. Uh, okay, so that's it for this problem. Thank you very much. I haven't.

In the question. The given different chili question is peace. Where? Why the world ash less six p. Y dash last six by a monster zero at big director than zero on the function. Given this FDA, it will still be square. No moving towards the solution. Why Dashwood bakewell toe you dash. Okay, do you to defy minus two minus to you be by speak. You on Dwyane. Never. Nash would be Puerto. You never Nash be be to the par minus two minus for you being by be you less six you plea by you to depart for now, putting these values And they given a question reword that peace where you have a rash being you to the par minus two minus for you, Nash B and cooked it to the par minus three. Last six. You be here for the par minus four. Last six, be your dash. Be going to the bar minus two minus. Will you be encrypted to the par minus three Last six. You de in country to the par minus two. It bolsters enough that will come out. Toby, you double Nash. They less to you. Dash p and complain to the bar minus one equals zero. No, no, you dash b equals w. That is very w by very is a little store. My next room w by see, that is you so well. It's too integration off Very w by the real, there s minus two Bt by b integration that we onda beautiful. You will require Tok upon he when? No then you by very people people. OK, bye. He's where that is. You would be Puerto integration off by piece where beating against minus a might be less seem so The solution for the given differential equation will be why it will still see one. Hey to the par minus two Unless they don't pay to depart minus three. Thank you.

Suppose we were given these two functions F of X and G fx. And if we wanted to evaluate f minus G of four, The first thing we need to do is figure out what that even means. What that means is that we want to evaluate f of four minus gee of four. So what does each of those mean? Well, F A four means that we're going to be using the F function to this one here and wherever we see an X we're gonna substitute in this four. So we would have four raised to the second power plus three. Let's see four square it is 16-plus 3 is 19. And then we would do the same thing with the g function. We would evaluate G of four which means that wherever we see an X. In this g function we're going to substitute in A four. So we would have negative four times so a negative two times four plus six. So negative two times four is negative. Eight plus six is negative two. Now, because this f of four equals 19-, and the GF four was negative, too. This is 19 minus a negative two, which is plus two, which would give us 21.

Suppose we are given these two functions F of X and G fx. And we wanted to evaluate F times G of four. And what that means is that they want us to do f of four Multiply that two G of four. So the first thing we need to do is to figure out if a four and GF four. So when we look at F of four, that means that we're going to be using the F function this one here and wherever we see an X. We're substituting in of four. So we would have 4 to the second power plus three and four times four is 16 plus three is 19. Yeah. And when we do G of four, we're going to do the same thing only. Now we're using the G function and wherever we see an X. We're substituting in of four. So we would have negative too. Times four plus six, so negative two times four is negative. Eight plus six would give us a negative too. Now we said that FA four was 19 And GF four was a negative, too, and 19 times a negative two would give us a negative 38.


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