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Following equations, find the most general function N(XY) so that the equation is exact For each of the [4y sin (Jxy) + dx + N(xy)dy = (b} [ye zx Ax6y + dx + N(x y)...

Question

Following equations, find the most general function N(XY) so that the equation is exact For each of the [4y sin (Jxy) + dx + N(xy)dy = (b} [ye zx Ax6y + dx + N(x y)dy (a) Selecl Ihe correct choice below and fll in the answer box t0 complete your choice. (Type an expression using and as Ihe variables g(y) where gly) is an arbitrary funclion of y h(x) where h(x) is an arbitrary funcllon of X

following equations, find the most general function N(XY) so that the equation is exact For each of the [4y sin (Jxy) + dx + N(xy)dy = (b} [ye zx Ax6y + dx + N(x y)dy (a) Selecl Ihe correct choice below and fll in the answer box t0 complete your choice. (Type an expression using and as Ihe variables g(y) where gly) is an arbitrary funclion of y h(x) where h(x) is an arbitrary funcllon of X



Answers

For each of the following equations, find the most general function $$M(x, y)$$ so that the equation is exact.

$$\begin{array}{l}{\text { (a) } M(x, y) d x+\left(\sec ^{2} y-x / y\right) d y=0} \\ {\text { (b) } M(x, y) d x+\left(\sin x \cos y-x y-e^{-y}\right) d y=0}\end{array}$$

So you had to find the most general function and off extra Mirwais. Such death, this equation. Why? Who signed off? That's why. Lost even eggs D X plus end of eggs. Come on. Why do I need quarters? Zero is exacts. So in this case, this is my I am so my am off Extra more? Why? Because why? Co sign of X y Lost to the ex Not implies that GM do. Why he quotes who used to put up with on these you have course sign of x y, then minus because we're different shit cause you have minus sign That's minus X y Sign of X y Now I want things to be equal to the end The x for this to be exact. So that implies that I want the end the eggs equal to co sign of x y minus X y Sign off, ex wife. So too soft for a what you do is just integrates both sides corner into ground on do derivative so well integrated besides off end off extra why he called to the integral of course. Sign of X Y minus X y Sign off X Y. I went into good D's because you are the eggs into good. This with respect to X. Now, um, what is I got into? Good piece before these? You have to use chin room now. Let me do something by the side. You called us into girl off you'd V records a UV minus into a god off being you. So for this case, I take my you to record two X So my do you equal to d X? And I said my DVD to be signed x y. So my view will be when it integrates Sign. I have minus co sign off X. Why, then I'm into good to inspect Your eggs would all be over. Why? So that implies that when I want it into good these off integrating consent of X y We're specked eggs. What big The integral of sine is science. I have signed off X y over. Why? Minus this is not you. My use eggs. Giddens minus co Sign of X Y. Why? Minors into grabbed off the my Viennese miners call Sign off X y over. Why them? I do You is the thanks and I have brackets closing these now This will be nothing Bad's numbers. My end off extra mile. Why is not in both? Sign off x. Why? Why we open that brackets or behalf plus x co sign of X y and they have the integral. Of course. Sinus is my Sanjay are minus sign x. Why? Why? And they are a plus g off. Why so when you do it is disconcerting days. So you have that and off x y equals x co Sign of X y lost gene off. What laugh of the bue pods we have. So in this case are am off extra. Why is why eat X y minus for X cube? Why lost soon? No, I am Do why equal to using production on these you need to the x y lost ex wife needed X y minus for X cubed. I want these to be equal soon Do n do X. So that implies Dad's Dubin. The eggs equal to eat X y lost its why eaten x y minus for X cubed too soft for and integrate both sides. You respect your ex so in cigarettes in both sides, both sides with respect to X. That's with respects so you have and off extra. Why records or the girl off easily? That's why. Loss X Why eaten X Y minus four x cubed Do X Now the way into good deeds you have you have e to the X y over. Why now to integrate these we restaurant to Ed's. Let me pull out Why? So I need to think about what is the into guard off ex eaten x y said. I mean after use chin room, which we just up there. So in that case, out sick. So I'm looking for the integral off X eats and it's why. So I take my you to be go to eggs, my Duveen, to go to e to the X y so that implies that my do you equal to the eggs. And then my V went into good district's respect. Eggs. You have eaten ex wife over. Why? So that implies that swimming into goods these off You're ving, which is X more brother Ethan X. Why, why minus one cigar, low key, which is eating the X y and I'm plus one over. Why then do eggs and then back to the days when into good disrespect to eggs out off minus extra before and then plus on each off. Why? Because I'm integrity. Respect to X. So this is just the call to eat the X y over. Why Lost? Why would cancel wise? I have X eaten. That's why then this while also consider these have minors when integrates e to the X y So we have these consist discusses gins. But when integrates eating the x y you I've eaten the ex wife Why? And then I have my minus excellent full and enough my each off Why discusses days. So that implies that my end off extra why he calls you ex eating. That's why minus x four most peach off. What?

We want to know which of the following functions are solutions City differentially freaking. So this is our front differential equation. So we want to find our wide double primed and our wives and put those in Syria. So let's start with a Tony too far and wide over prime. So first, let's remember that the derivative Ah, cause, uh is negative, son. And the derivative of Sign is positive co sign X. So just some odd interviews. Do you remember? So let's take artery the sign we're gonna get Coast Onyx and I protected a room that we get negative sign. It's so we're just gonna put those into our equation here so we'll have negative sign picks plus sign X people to sign X and more left and saw. This is zero, and it's a sign X These are not equal. Therefore, we know A is not our answer for bee. We're under the same thing to define what crime, which is negative. Signed X and then we're gonna get our wide double product, which would be bring out our negative review and sign is co sign so negative co sign it. Put those into our equation, will have negative co signed X Please Costa on X, you bring down our right side. Far left side of zero will know Right side is the sine X and these are not the same. Therefore be Zlata. Answer Joyce Fighter Chrissy, we need to find what crimes in one double prime placement or equation. So it's Yvonne. Why prime? We're going to use product will. So how was the first times that a review of the second close? The circuit times the derivative with first we said before that will have one have that. Oh sonic close one have signed it. We're gonna do the same thing to find my double prime. We have to use product rule for this portion here. So we say the first times that ever got a second plus the second times that during the first it's just one upside and they were students. Bring down this portion here, but it's derivative and not derivative is one have co sign it. So we simplify. This will have negative one happy ex tonics plus 1/2 co sign IX. That's one of those on exceptions one ex villages, but no one here or understood one. There So now we have We're why don't probably equation and r y equation and you just plug those into our function. We'll have negative 1/2 ex Sonics, close coast onyx lists or while equation and bring down our right hand side. So we simplify. We have a negative 1/2 X I next year on a positive here. So just counsel with each other and we have co sign X equals sine X over. This is not true. Therefore, seize letter answer toys either Does it look a Shame day? So even though we know that this has been an answer toys, we could still work it out. So we have our wives functions. I need to find our wives crime. So it's right there. You say the first times the ribbon of the second plus the spec It times that you're one of the first possible bodice will have positive 1/2 explore in it. Wilderness won her goes on it. Never use this upon our war double from equation. So take the first times that removed a second plus the second times that during your with the first and I was the derivative. This portion. Now it's like through this portion. So we have a minus one have sine X, but it changes. Sign because they removed could sign in the negative sign. But we simplify this. We'll have a one have ex co signed X and then just be one happens when half that's just one. So but signing? So now we have off the equation. So we need to plug into our differential equation here. Threw in a pregnancy. So you have want, huh? Thanks. Co sign it. Let's find it. Plus a negative one, huh? No. Co sign it and bring down our right hand side. Definitely. Look here and here. These subtract out they were left with signed X is able to sign it. Therefore, D is our answer choice.

In this video, we're going to be looking at verifying these solutions to a differential equation. So the differential equation we are given is why double prime plus Y equals sine of X. And we have a variety of solutions that we want to check um as to whether or not they are solutions to the differential equation. So let's start with this one here. Why equals the sin of X. So we need to plug this into our differential equation. We know why, but we need to find Y double prime. So let's go ahead and do that. Let's take the first derivative to find why prime first and the derivative of sine of X is cosine of X. Then if we take the derivative again to get y double prime, you will get negative side effects. So we plug in this here and it's here. What we will end up with is negative sine of X plus sine of X equals sine of X. So our left hand side simplifies is zero. So we get zero equals sine of X. And this is not always a true statement. So we cannot say that this is um necessarily a solution. Let's move on to be. So this is the function we're giving. Let's find Y double prime. So why prime is going to be negative sine of X, Y double prime. Going to be negative cosign effect. So we plug that and we get negative co sign effects plus co sign of X equals sign effects. So again, we end up with zero equals sine of X. And this is not a true statement for all values of X. So we cannot say that this is a solution to our differential equation. Moving on to this next one. Let's um check this. So why prime have to use the product rule here? So x over two co sign of X um plus 1/2 sine of X. It's going to be our first derivative and then our second derivative will be the derivative of that. So again we have to use the product rule on uh this first term here. So we end up with X over two negative X. Over to you. Sign effects plus one over to co sign effects plus the derivative of this second term, which is simply one over to cosign. So that simplifies a little bit. We have Y double prime equals negative effects over to sign effects plus. Cool. Cinemax now let's go ahead and plug that into our differential equation which was again white girlfriend plus Y equals sine of X. So negative X over two sine of X plus. Co sign of X. Um Plus are why? Which is in this case. Thanks are to Cinemax equals sine of X. Um So these two terms cancel. And we're left with co side of X equals sign effects. Which is again not true for all values of X. So we cannot say that this is a solution. And now let's try our last function. Why prime going to equal if you use the product rule again negative X. Uh Sorry, this will be Positive x over two sine of X right -1/2. Co sign of X for our first derivative. And then if we take the second interpretive it should be just literate, first derivative using the product rule. Again, we are going to end up with X. O. Or to co sign of X plus 1/2 sine of X minus the derivative of this year, which will just be was one half sine of X. So again you can see these two terms can be added together. Get X over to co sign of X plus the sine of X. Now let's go ahead and plug back into our differential equation. So X over. Just sorry, X over to cosign effects plus sine of X plus negative X over to co sign of X equals the sine of X. So these two terms on the left hand side cancel out. And we're left with sine of X equals sine of X. And that is true for all the effects. So we know that um D is the only function networks as a solution for all values back.

In this problem, we will cover computing partial derivatives. So we want to find the partial derivatives of respect to X and Y. Of each of the four expressions given and blue. And we will plug them into the equation written in green to see it if they satisfy it. So we will begin with the expression X squared times. Why? Cute? And we want to find the partial derivative perspective X. And this means we're going to hold why fixed. So we're treating it like a constant. So we really just want the derivative with respect to X. X squared while keeping Y cubed the same. And this yields us two x. Why huge? Now we want to find the partial derivative respects why? And now this time we're going to hold X fixed. So treat elected constant. So we're moving it to the front and we want to find the partial derivative expects why of why. Cute. And that yields us three X squared Y squared. And if we were to plug this in to the equation given to see if it satisfies it. We will see that the left hand side would not equal the right hand side Because the left hand side would equal five x squared. Why cube? Which is not equal to explain why cube. So this expression X squared, y, cube does not satisfy the green equation. So now we move on to our second expression which is X plus Y plus one. And we want to find the partial derivative perspective X. So we're going to hold Y what uh fixed. So we're treating it look at constant. And that means we're really only looking at the X. Turn Which gives us one is our partial derivative. And now to find the partial derivative perspective, why? So we're holding X fix now treating it like a constant. So we're really only going to look at The wide term and that gives us one as well. And if we plug it in to the equation in green we will get that. The left hand side is not equal the right hand side. So that means this expression in blue does not satisfy the equation in green. So now we move on to our third expression which is X squared plus y squared. To find the partial derivative respect to X. We hold why fix treat like constant. So we're really only looking at the derivative of X squared in this yields us two X. Find the partial derivative perspective. Why? Now we are going to hold Xbox and treated like constant. So we're really only looking at the white term which is why squared. And that yields us to why? Now if we were to plug this into the equation in green, we will get that. The left hand side does not equal the right hand side because we will get two X squared plus two Y squared. And that does not equal our original function. So this expression does not satisfy the equation in green. And finally we move on to our last expression and to find the partial derivative with respect to X. We are going to treat why like a constant. So we're looking at X to the .4 while keeping the wide term fixed. And that's going to give us 0.4 X to the negative 0.6 times y to the 0.6. Now we want the partial derivative with respect to Y. So that means treating X like a constant So that X to the .4 is going to stay the same While we want the derivative straight. So why why to the 0.6? And that will give us 0.6 x. to the .4 times Y. To the negative 0.4. And if we were to plug this into the equation in green, we will see that the left hand side does in fact equal the right hand side. And I will show this in a second. Oh my bed, that's supposed to be a .4. So we get that First. We have 0.4 X. To the fourth Wide to the .6 Plus 0.6 X. to the 4th times. Why? So the 0.6. And because because That 0.4 and 0.6 add up to one. We get that both sides equal each other. So our last expression Extra .4 times why to the .6 satisfies the equation in green. That means d. is our answer.


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