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ETERCISE 1 Show that the set of all solutions to the equation 3y ; 2 € (-x,x) = given by the tTO parameter family of functions {yab}_xsSso where if CI < r ...

Question

ETERCISE 1 Show that the set of all solutions to the equation 3y ; 2 € (-x,x) = given by the tTO parameter family of functions {yab}_xsSso where if CI < r < 0 Yc,b if a <r <6 if 6 < x < x if CX <r <6 y-x. if b < r < x if CI < r < 0 Ya.> if a <r < xy-xx

ETERCISE 1 Show that the set of all solutions to the equation 3y ; 2 € (-x,x) = given by the tTO parameter family of functions {yab}_xsSso where if CI < r < 0 Yc,b if a <r <6 if 6 < x < x if CX <r <6 y-x. if b < r < x if CI < r < 0 Ya.> if a <r < x y-xx



Answers

(a) For what values of $r$ does the function $y=e^{r x}$ satisfy
the differential equation $2 y^{\prime \prime}+y^{\prime}-y=0 ?$
(b) If $r_{1}$ and $r_{2}$ are the values of $r$ that you found in part (a)
show that every member of the family of functions
$y=a e^{r_{1} x}+b e^{r_{2} x}$ is also a solution.

This lesson in this lesson, we have the differential question here. Um The family of solutions. So we would verify F. This is actually a solution to the differential equation that we have. So we will look at the left hand side, the right hand side. So we have Y. And will differentiate the two times. So the first different show of why eggs the whole of this. If we are differentiating that we have to see one E. Two X. Then class. This one has a proud that. So we hold one and differentiate the other. For the first time we would hold C. Two X. When you differentiate into the bath to actually would have to see two X. The first step, the second one is when we hold you to the about to extend differentiate C. Two X. M. By the C. Two E two X. All right. So this is the first differential. That second differential is taking the whole of the first differential and differentiating it'd gain. If we differentiate this, we differentiated timely. The first time we have 4 C1 E two X. The blasters. We differentiated us up a product that we have four C two X E two. Then we differentiate the other side. That gives us to see to E to X. Then we differentiate this one. We have two C two, E two X. Okay, so let's look at those of that common and add them. We have this so four C one E two X. Last four C two E. Then that is four C two E two exploit. So we can place them in here to see if it's equal to the left hand side that is zero. The right hand side is there? So we have the second differential, the call of four C one E two X plus four C two X E two X. The last four C two E two X minus four times the first The French Shell. So four times the first different show if it is a friend show is to C one E two for the last two. C. Two X. The last C two. E. Two X. Okay. Done. Plus why? For why? So? No. You know y. s. c. one e. 2. S. Well I see. Two X. E. Two X. Okay. So would look out all of them together then. Oh so long. Okay. So let's do with them based on the ones with X. And the ones without X. Oh okay. Okay. The best thing to do is to go into all of these. Then you factor out E. To the power to X. So here we can factor out four. So the E. F. C. One. Yes. Then we have C. To about there C. Two X. Then we have C. Two over there. Right factored out the four. Yes. E. To the about two. So here we have C. One E. F. C. Two X. And here you have C. Two. All right, let's look at the second one. You can factor out negative four E. Two X. A little bit left with to see one then plus to see two X. Then uh Plus C. Two. Okay. The last that we factor out four E. Two X. Came Then we have seen one plus c. two. Okay, so out of that we can still fucker out four E. To the path to so that all of that can come into a big bracket. So here would have C. One last C. Two X. Last C. Two. Then all of these would become negative. So you have negative to see one -2 C. two x. The negative C. To all of that would be negative. Then last but not least we have what we have here 12 So that is C. One plus C. Two X. All right. So we can now look at them together. We fc one Than negative to see one and see one. So this becomes to see one minus two. C. One. Alright. The next thing that will look at A C. Two X. So you have C. To xia negative to see 2x. Then you have to see to a sugar becomes to see two X minus two C. Two X. Then the last step we have C. Two here then a negative C. Two. So this week um si tu minus C. Two. Okay so fall out and as we can see this is the exact upsets of each other. There's that upset of each other. Is that upset? So that becomes very necessary. So the left hand side is there, Let's go to the right hand side. Is he called the 0? All right. So that is actually the family of solutions for the differential reposition. Now let's look at the interval. It goes is a polynomial. Uh so it goes from uh the logarithm function that is multiplied with X. So it goes from negative infinity to positive committee. Okay, so thanks for the time. This is the end of the lesson.

We want to find values of ours that satisfy you. Your parents really crazy to our fine place. Wide time on his wife. So first, the you always wanna figure out what values presidency, there's a place. So, you know, we're gonna need a wider Wilpon and a wife. Son, we already have our wife. So let's burn our Watson, please. So the river Look, this is just our son to the Rx. Using changeable. You talk to the reforms of what we do inside there and explain it, which is our This was a part of you. Now we need to burn. Why double clutch? So it's important to remember that our is a constant because we're still using chain will not product, Lou. So is there gonna use tangle here? So this is just the derivative of dysfunction. Witches are some of the most fun. Are times are, and then bring down the use of our ex. Put a simple pas into our square YouTube oryx. That's why it's important to remember that that which table there is a constant there is otherwise you would find the spot of fools, but that not the right way. So now we have all their bar functions and upload them into our original in question. So we have to plugging are wide open. Searchers are square inside or it first war prime, which is are times into the arcs and Linus Arsenal equation witches use of our and I want a clear that is equal zero. So Lynn distribute are too. So I would become too are square you to guard it. Plus, are you to the O r monitor Sting to the Arctic Sea Close here. There we see it comin back there. You need to know our exit, all the memories. So we're gonna And when we do that, we're left with two r squared. It was our Mona's warrant equals zero. So when the only factor we need you to the oryx equal zero And we also need two times r squared butts are minus one equals here. No, our exponential functions never equals in Europe. So we can scratch that. We can eliminate that. But now we're just looking for a win to r squared plus R minus one in the room and it makes you more comfortable. You replace ours with exes, but essentially, it's this thing. Someone a factor by greeting. So are you that were you multiple are the He's to get negative too. There wasn't no adds a positive force. So we're back to the two are one and two. And you have to make one of these negative toe add. It's a positive one. So are one has been a year. So we're back here by greeting. You just put up our little turn. So we have two ex cleared to it minus one, and then bring down on minus one. We're gonna group based in tires and they want a group based in turn. And we're back throughout the greatest Coleman vaccine. The greatest benefactor of used to it. Exposed one. They're gonna bring out a minus one here. Well, let's put explosive forming on the inside. Parts and factors are two X minus one and X plus one. Peoples here again Here. I'm going to have I'm going to set both Are factors equal to linger. So we are two x minus one equals zero anyone X plus one close to Europe. So here we're gonna add one of both sides. Do you have to ex people, boy divide by two x equals 1/2. And here we just attacked one from both sides of X equals negative one. So we found our to our variables. But what values of our So I wanna write this down here are equals one and negative warns Those are two are values that satisfies the operation that we saw fourth year So part b ext for us to use those values and so that every member of the family of functions of this new equation here also satisfies this equation here. So again, we're gonna be We're gonna need a wider with Parma. What? Klein and our regional war inflation. So we know we're gonna find our Why permanent crazy? No, I don't want money, but the first up is deployed in an hour. Variables are are variables. So you know that why equals very times being to the 1/2 X exploded in our way was being from speaking to the negative one exam's gonna the negative me about the wood. So we have our original. Why places now we need our weren't hard equations, so that's like the river. This is so this is the constant. So we just started a review of our exponents, which is 1/2 terms first using the one helping in something here. He's a constant. So just like that, the resistance of our upper selection, which is negative forms, I just just minus the negativity. And lastly, we need are wide open. So again, we're in the river with this up here because you both the constant no one have terms. One, huh? A. You need to know what happened you're doing We're gonna take our derivative of this inner bumpkin because that's called since Bill. So I don't gotta become plus, being to the negative. Yeah, a negative times. But something about it is we have 1/4 a. You need to go in half X 1st 3 times needs binning the fix. And now we have all three of our functions of the wooden tur equations. Here. I want you to set up a way we're gonna pull you everything. Are we up to times our wire double burn equation? A little pleasure in one fort area In terms, beings of the 1/2 X plus being times use of negative bricks Bus are wired for inflation, which is one have a use of the one, huh? Ex minus the times you to wait a bit. We wanna plug in our worry equation. Yeah. Don't forget to say that people here, the grandest of you, So have one. Have every times you need to be one. But first to be what? And then just bring everything else down them something else to distribute. And we're going to be this negative here. So, um, honest Eddie turns into the 1/2 ex. Let us be you with here. So now we have a bunch of variables and numbers, so we're just gonna simplify. They stall, interfere the basis. So I know there wouldn't combine news through. So it has a That's just one. So what a times need to the 1/2 ex gobbling up with 1/2 of one. Uh, next week. Just bring down this Your so for us to be using, Linda. But the family can bring the sound here. Some honest. You, you need to read it. You could go behind those local bar news. So that's two minus one is just give us a positive or in here. So have they close the terms means and negative six and then lastly, you're gonna bring this down. So we have a minus Any times Easy. The one happened. No, we can bring that down to the finest being used for Magnetics below zero. And as you see, you have a positive and negative those council and a positive and a negative. So those council, so zero does equal zero. Therefore, we verify.

Here. We're trying to verify that this why is a solution to this differential equation. Uh To make things a little easier, we can rewrite all of these derivatives as you know, their prime format. So this one is going to be why triple prime. This one is going to be Y double prime and this one is just gonna be white prank. So I need to take the derivative of this one and take the derivative again and then take the derivative again to get my third derivative and then plug everybody into the equation and verify that it is true. So first things first, I'm going to find this flight prime. So all these seas are constant. So taking the derivative of that, I bring the negative down so negative C one X. To the negative to power Taking the derivative here. I just get c. two taking the derivative here is a product rule. So I need to do it in pieces. So I'll take my c. three out here, Derivative of the first part is one times natural log of X. So L. N. X. And then plus I leave the X alone and multiply by the derivative natural log which is just one over X. And then plus the derivative of this bit. So that's going to be eight X. I can uh let's go ahead and simplify that first derivative just to make our lives easier. So I got negative C. One X. To the negative two. And then I've got to see to um and then here X times one over X is just one. So I'll write that down is one right there And I'll distribute the c. three. So I've got C three times the national log of X. And then plus c. three. And then I have my eight x over here. Now let's take the second derivative. Yeah. Okay. Take the derivative of this thing and the negative two comes down negative times negative is positive. So to see one X. to the -3 power. The derivative of the constant zero. The derivative of natural log of X. S. One over X. So I have C. Three times here. I have C. Three times X. To the negative one power Instead of one over X. Just make it easier for the next derivative. The derivative of C three is zero and then the derivative of eight X is just eight. Don't need to do any simplifying there. So let's just find the third derivative and then plug everybody in. So 123 carry the negative three down. We get negative six. See one X. to the -4 power take the derivative here, take that negative one down. So I get negative C. Three X. To the negative to power. And then the derivative of eight is just zero. So let's plug everybody in where they go. So I've got change colors X cubed times the third derivative. Which is this last one here. So negative six. See one X To the negative 4 -4 C. Three X. To the negative too. And then plus two X squared. So Plus two x squared times The 2nd Derivative 2nd derivative is right here. So I've got to see one X. to the negative three plus C. Three X. to the negative one plus eight. And then scrolling up, I go minus X times Y. Prime. So minus X. Why prime was all this stuff here. See if we can fit it. So negative C one X. To the negative to power plus C. Two plus C. Three natural log of X plus C. Three plus eight X. I'll make sure all that's in the same parentheses there and then we have plus why and why is what we were given. So just carried down here Plus why? Which is mm. See if we can prove that. I know we can't C. One X. To the negative one. Power power C. Two X. Plus C. Three X. Natural log of X plus for X squared. And that's and all of that should equal 12 X squared. Well let's distribute what we can and then simplify as best we can. So let's take this ex gave it and put it with everybody in there. So we've got negative six C. One X. To the negative one. Power three minus four Then -C. three X. To the negative one. Power no not negative was 3 -2 is one. So just positive £1. Then we distribute this two X squared to all of those guys two times 2 is four and then see one and then X to the negative one. Power distribute again plus two C three X. to the 2 -1 is one Plus two times 8 of 16. So 16 X square. Now let's distribute the negative X negative times negative is positive. So plus see one X To the -1 power plus or sorry minus C. Two X, distribute again. We get minus C. three X. Natural Log of X. Take the ax and put it over here now down minus C three X. And then distribute it one more time minus eight X. Square. Okay? And then I'm just gonna rewrite all that stuff down there. So that's uh plus C. One X. To the negative one power Plus c. two x plus C. Three X. Natural log of X plus for X squared. And hopefully all of that is equal to 12 X. Let's see what we can do. All right, let's look at what we got. Let's look for all the negative one X. To the negative one power. So there's one here there's one here there's one there There's one there and I think that's all of them and they all have AC1 with them. So they're all like terms. So we've got a negative six plus four. That's negative too. Plus one negative one plus one again zero. So all of those cancel out now let's find everything with just an X. In it. So here's an X. Here's an X. Here's an X. Um minimum. Here's an X. And here's an X. Now not all of these have the same sees. So these two have c. three. So that's negative to see three X. Uh And then a positive to see three X. Of those all cancel each other out. Then I've got a C. Two X. And a negative C. Two X. And they cancel each other out. Excellent. Um Let's I'm gonna leave the X. Squared because I wanted to be the last thing we do. So let's look for the X. Natural log X. Is there another one in here? Yes. And it's a negative C. Three X. Natural log X. And a positive C. Three X. Natural log X. So they cancel each other out. Now I'm down to my uh X. Squared. So uh 16 minus eight eight And then plus four is 12. So we are verified.


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