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Select all the exact differential equations In the following Ilstty 'ty dyldt =Y sir(t) {cos(y) dy/dr =Zyt + (+1) dyfdt =t dt +Y dy = 0Y dt + [ dy =cost) (sini...

Question

Select all the exact differential equations In the following Ilstty 'ty dyldt =Y sir(t) {cos(y) dy/dr =Zyt + (+1) dyfdt =t dt +Y dy = 0Y dt + [ dy =cost) (siniy) dyldi =Ly dt [y dy =Y dr - ! dy = 0 cosit y) tcosity} dylot = 0cositV cos(ty) cy/dt =

Select all the exact differential equations In the following Ilst ty 'ty dyldt = Y sir(t) {cos(y) dy/dr = Zyt + (+1) dyfdt = t dt +Y dy = 0 Y dt + [ dy = cost) (siniy) dyldi = Ly dt [y dy = Y dr - ! dy = 0 cosit y) tcosity} dylot = 0 cositV cos(ty) cy/dt =



Answers

Find the general solution for each differential equation. Verify that each solution satisfies the original differential equation.
$$\frac{d y}{d x}=\frac{y}{x}, x>0$$

De y over the X equals Y over X and also a Delaine restriction here. X has to be greater than zero. That's important for this one, because if X equals zero noticed that this differential would be undefined. So it's a good thing to have labelled in a lot of times it's given to us somewhere in, like, the original question. So just be mindful of that. If I want to get rid of the differential, I need to take the derivative or sorry anti derivative, which would mean I need to separate the variables so I would move that DX over to the right hand side so D y equals y over X times DX. But then notice on the right hand side. Now I have X's and y's so I need to move this Why as well since this wise on the new reader and have to divide it over to the other side. So I have one over. Why do y equals one over X DX, right? That ex needs to stay on the denominator where it was originally. So at this point I'm capable of integrating. I'm going to the integral on both sides, keeping it balanced. And both in a girls are of the form one over you. Do you? So this is unknown anti derivatives unknown in a girl set up and its anti derivative is Ln of you It's a natural log of you. So on both sides, I'm actually use that natural log set up have Elena y and l N of X. And then I put plus C because I don't know what constant number was added there. So at this point, I'm going Teoh, use e to cancel out that Ln I won't cancel it yet. On the right hand side. Watch how that happens. So e and Eleanor in versus. But I need to pause for a moment here before I try and cancel it on the right because that whole thing becomes the ex. Phone it. I can't just take it term by term. Since the exponents are adding, the bases of the exponents are multiplying. This is where it came from, like before those exponents were added together. Right? That's the algebra of it. We know that if we think about variables like X times X squared, you wouldn't even blink. You'd say x cubed because When the bases are multiplying, you add the exponents, you just do it automatically. We'll hear the exponents or adding, so the bases need to be multiplying. Okay, that gives me the power to make this term this e to the C actually into just one big constant term. I can write it as one big, constant term because it's constant. He is 2.7 ish, and C is just a constant number. So that would just be some set value. My calculator. It's not gonna change. It's not a variable, so I can just replace it with still a constant now that I have those terms separated E. And Eleanor in versus So Actually, the simplest way to write this as that anti derivative, as that general solution would be to write, Why equals I'll use my capital. C Still see Time's X

Problem town we got. Why? Square months wine But lying to you Why Overpay Axe? He was Jack's way. Stand you to separate a barb wire. We just need to move by the ex those side. That's kind of what square minus y t Why equals X t acts and were integral Both side his fart. We got Juan certain Why Cube? Linus, one half y square equals this partisan want have at square And don't forget a constant So does our solution. And if you want to check, differentiate beside with respect Why what? Why square to your wife? The axe So track. Why do you buy t ax? Those two of ours people packs so exactly dysfunction Because this problem ten.

Consider the different calculation. Why squared minus y times the wire, p X equals X. If I want to get rid of that desire, Buddy X, I want to do the integral anti derivative and then that's gonna give me the general solution. So in order to do the inner girls, I need the variables to be separate with all of the wise on one side and then all of the exes another. So I'm going to multiply the D X over to the right hand side. So that way I just have wise over here on the left and I'll have just X is on the right. Then I'm ready to integrate because I have the variable separate. So I do. The integral is it's the anti derivative on the left. We're gonna go turn my term. I need to add one to the exponents and then divide by the new So 1/3 Why cubed minus 1/2 y squared. And then on the right side I would get 1/2 X squared. That's gonna ugly. And then plus C, I write plus C because I don't know what constant number is there, but I know there's some constant over there. It might be zero or might be any other constant number. So I put plus C to hold its place, and that makes it the general form. Now, this is actually an acceptable answer, depending on what your directions might say. But a lot of times we like to get rid of the fractions or get rid of coefficient in front of the wise. So, like I would multiply everything by three to get rid of that 1/3 and notice here I just write C. I don't need to write three C because C is just some constant. I don't need to say this is three times just some constant. I don't know. I still don't know what it is, so we just keep calling it. See, you don't have to relabel it. And then I would actually fly by, too, as well, because I'd like to get rid of that 1/2 fraction so multiplying by two to get rid of those 1/2 fractions, I would have to why cubed minus three y squared equals three x squared plus C. And again I could just right see, because it's still just some unknown constant I don't need to say I doubled some unknown constant because it's not like it's supposed to be fixed value right now. I haven't labeled it or given it any value, so I could just leave it still unknown.

I have d Y over the X equals e to the y squared all over why this is a different Joe equation, because I have that d y over DX. And if I want to get it back to a general solution that I need to do the integral. But first, I would want to separate the variables so I would send the d X over to the right side because that gets it off the denominator. But I noticed that really have the wrong variable there with it. So I'm gonna need to move this entire rational over to the left side. An easy way to do that is to use the reciprocal. So if I use the reciprocal, that's me flipping that fraction so that this will cancel out so that reciprocal to keep things balanced goes over to the left side. So my new line of work is gonna be why over e to the y squared times d y equals just DX or could write one DX Now that my variables air separated, I'm ready to integrate integrating on the left hand side. I want to do one important step first. There's not like a great rule for having an e on the denominator. So I actually can bring that e up to the numerator. If I write, the exponent is negative, right? That will reciprocate it for me. So I'm gonna write that exploded as a negative y squared to bring it up, and everything else is going to stay the same. And now becomes more obvious to me that I can use a substitution here to make it just e to the U, which its anti derivative states the same. It's still eating the you. So we're going to replace this exponents negative y squared, and I need to also check its derivative. Well, the derivative would be negative to why d y and I have some of that already here notice I have. Why do you Why? So if I solve for what I already have, then I would need to divide that negative to to the other shy. So as we're place, everything we have each of the U. And when I seven for this, Why d y I get do you over negative too. Okay. So again, I saw for what I have currently in my integral, which is why in D Y, which meant I had to move that negative to over that negative two is out of place and then replace the you that I saw it for and replace this. Why do you? Why now it's ready to integrate this negative two on the denominator is just a coefficient, and the anti derivative eat of the U is eat of the U. But let's change it back to our current variable or the correct variable. And so we're gonna call it negative 1/2 e to the negative Y squared on the left hand side. I'm sorry. On the right hand side, there's not a whole lot of work cause it's one DX wine goes Toe X. When I do the anti derivative, I need to have a plus C. I put plus C because of the fact that there's could be some constant there that I don't see when I took the derivative, because any concert number has a derivative zero. So just be careful when we have these indefinite integral to always input plus C plus C plus c. So just to make this a little bit cleaner for my final answer, I would get rid of the fraction in front. I would multiply by negative too. But notice I keep the see the same because, see, it's just some number I don't know and they have two times Some number I don't know is actually still some number, I don't know. So I just leave it as a C there at the end, and this is a more simplified version of the general solution.


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