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Numerade Educator

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Problem 8 Medium Difficulty

Solve the differential equation.
$ \frac {dH}{dR} = \frac {RH^2 \sqrt{1 + R^2}}{\ln H} $

Answer

$-\frac{\ln H}{H}-\frac{1}{H}=\frac{1}{3}\left(1+R^{2}\right)^{3 / 2}+C$

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Video Transcript

All right. So we're gonna solve the differential equation. And this one is a pretty simple ah, separable differential equation, I would say so we noticed that we have two variables. So, uh, I'm gonna start off by multiplying d r and l eight herb natural log of h people double sides. So we're gonna have natural log H g h equals are each where one plus r squared tr and we're also going to divide this beach squared by both sides. We're gonna have l and H over h squared th equals r squared D Earth. It's never ending up. We ended up with this, and since they both have but D h where each side has de and their variable, that means that this here is a derivative of something. So we're going to find the net up. We're gonna integrate both sides. I'm gonna write the same thing, but with an integral good. All right, so I'm gonna start a put this side, and then next time, I'll do this side. So for this, and I'm gonna actually integrate by parts that we're gonna have you do you and then the maybe so integration by parts is you, b equals you. Be two girl. Uh, me. You already. So now that we have that, uh, I'm gonna Yeah. Okay. So I'm gonna Oh, no weapon. Here we go. I'm gonna plug in parts of this here, so I'm actually gonna rewrite this as, uh, inter girl and h times one over each square, which is th which is the same thing is disappear. So, for you, I'm actually gonna put Phil and h So the derivative that would be one of her age. Something very simple, Uh, for Devi since I need you and DB, which would be l and H and one over each square. So I'm gonna put one over h squared for D V. And then basically, I'm gonna find the integral of this here, which is going to be negative one. All right, so then we're gonna use these here. The plug in said this equation here, you're gonna have Ellen each times negative one over age minus into a girl of me, which is negative one over which times do you? Which is gonna be? They were going to be left off with negative l and H over H minus into grow of negative one over each. We'll just do negative one over each sport. D h, this would also be age every And that's just because this is th is. Well, yes, maybe already. Now, I have this never gonna have to solve this year. This one is really to heart. I'll go over this one, since I really didn't explain too much about how I got this year. So I will be right. This has negative each the negative two. Since one over h is equal to each one. This just means that it's the reciprocal. So me of this here, I'm gonna work on this. So we have you have hte needed to and to do the integral we're gonna add one and then put it over. Negative two plus one. So we have, uh, 1/3. No, just kidding. That's not right. Negative one times negative h with negative one. So these will cancel out. So then we will have one over each then. So that's just for this part here. You have negative, Ellen H over H minus one, Over. So then we're going to get this part over here. She's gonna copy and paste it if I can, man hurry. But here could make this equal to Can I pay some pre shirts? Yes, already. So we're gonna work on this end. This end is actually a lot easier because you don't have to use integration by parts, so we can actually just use ah, use up. So I'm gonna use u is equal to one plus r squared. Do you is equal to R D R. And then we're gonna have d R. Is equal to do you over to our So that means we can, but the's so then we have are over to our you do you let me just these cancel out, which is good, because we have the do. You shouldn't have one variable. So we called the two on the outsides. We have 1/2 into girl view, do you? We can also rewrite this is one over to you to the one half and then do the whole plus to choose nos. And then we're gonna have this is gonna be 3/2. So then we're gonna multiply this by their Super Bowl to over three. You prove it, You We'll see. So since we have this. We're gonna actually swap this. Well, What? We originally took out one flys are square. It's been a 12 to third. One close R squared 3/2. Let's see, you can both blood this across or I guess these cancel out. So it's actually one third one plus R squared three over to put seat. Not ready. So now we can have these equal to each other. So he have negative melon h over H minus one over age equals 1/3 one plus r squared three half. Let's see, the reason why we don't put pussy on both sides is because of milk cans each other out. So we want to have this plus C on one side, and that's how you should get this answer.