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(a) For what values of $ r $ does the function $ y = e^{rx} $ satisfy the differential equation $ 2y^{"} + y^{'} - 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 = ae^{r_1{x}} + be^{r_2{x}} $ is also a solution.

(a) $$\Rightarrow r=\frac{1}{2} \text { or }-1$$

(b) $$=0=\mathrm{RHS}$$

Differential Equations

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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.

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Differential Equations