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Solve ( xy + y2 ) dy +y2 dx = 0...

Question

Solve ( xy + y2 ) dy +y2 dx = 0

Solve ( xy + y2 ) dy + y2 dx = 0



Answers

Solve the differential equation.
$ \frac {dy}{dx} = x \sqrt y $

In order to get this equation into a form where we can see our P a vaccine or to determine the interpreting factor, we must first divide both sides. In other words, all the terms by the coefficient of y prop, which in this context we know is X squared Plus one. Do this and we get d Y over jacks plus three acts. Divide by ax squared plus one times. Why is three acts again? Divide by X squared, plus one. Okay, Now, or e to the integral of P three acts over X squared plus one D axe. Integrate this. Remember each the natural of X is one. We simply get ax squared, plus one to the power off three divided by two. I'm writing. This is a fractions. It's easier to read. Now. We must multiply all part of the differential equation. So all the terms by or integrating factor and then remember, this is the integral gonna be intruding the right hand side in a second. Okay, getting this, we now get the fact that we can integrate the right hand side. As I said, Okay, we know we can pull out a 1/2 right This is the integral of the squirt of you. Do you know what? Let's say that you is equivalent to X squared plus one. Do you substitution? Right now d'you is two acts DX 1/2 do you is equivalent to x d x. This is U substitution right here, which gives us 1/3 you to the three over two plus c back substitute in What? The value of us. Okay, cool. We have that. Now we know we have. Why is one remember? We want this in terms of why plus c divided by ax squared plus one. The three over to remember original initial value was wives zeros to simply plug in acts. And why, in order to solve for C, the sea is pretty straightforward. Simply one final solution. Plug in our seed into the equation We found out in a previous side and we have our solution

This question asked us to solve the differential equation three. X squared y squared Not what we know is that if this is r d y o ver de axe than r D y over, why squared is three x squared DX. This allows us to get the X is on the same side and the wise on the same side. No, let's take the integral. The integral of this is negative one over. Why be integral off this increased exports by one divide by the new exponents is execute. Don't forget our constant of integration plus C now lastly to right this just in terms of why we get wise negative one divided by X cubed plus sissy

In this problem, we have to solve the differential equation. Why? Double pane plus two y is equal to zero. Now we can associate the characteristic a creation. Tow this to function invasion which will be given by t squared plus two. Is it called Tuxedo? Now let's recall the Purim when we have the complex route off, the characteristic decoration puts this. So this is theory and 11 from the textbook. If the roots looks like Alpha plus I beata on our two is a call to Alfa minus. I better are the routes off the characteristic in creation, then the solutions or in general, solution is going to look like Why is he called to Ito the poor Al Fix times. See one times co sign off Beat X plus Tito time Sign off, be takes. So we're going to use this to solve this problem. So now D squared. Is it going to negative too? This implies TZ called too plus minus square root off I So that means my Allah phrase he called to zero and beat Isaac called The Squire rode off to so then the general solution off this differential equation. It's going to be. Alfa is zero and Ito Depot zero is indeed, it's a call to one. So see one times Co sign beater betide square root off two times six plus C. Two time sign off. Squire rode off to X, so there's going to be our final answer.

So for this problem, what we're gonna want to do is implicit differentiation and implicit differentiation is very good when we have some expression with, you know, an X and A Y, and it's equal to another expression. So we have this equation. Um, that's not just a simple asses. Why equals something? It's some function of X and Y perhaps and then another function of X and Y on the other side. So that's why we want to use implicit differentiation. So what we're given here is thesis ein of X y being equal to the co sign of X plus y. We're gonna differentiate both sides with respect tax. So it's gonna look something like this. So now when we do this, what we end up getting is we want to do the chain role. So, with the chain role we now have that this is the co sign of X y times. Why plus X y prime and that's doing chain role. That's gonna be equal to a negative sign of X plus y times one plus why prime? So with all of this, we want to simplify further. So we have now is why cosine X y plus x co sign X y times y prime And that's gonna be equal to the negative sign of X plus y minus the sign of X y x plus y white prime. So now that we have all this, we want toe ad, um, this right here to both sides and then we want to factor out the why prime. So when we do that, we get, um, X curse Sane X y plus sign X plus y We moved this over here and we moved this over here. So now that we have that, this is going to be all times. Why prime? And it's equal to a negative sign. X plus y minus. Why co sign X y and then we just divide the whole thing by this portion right here. So what we end up getting is that why prime is equal to a negative sign? Plus why, minus why could sign ex wife all over x co sign X y plus sign X plus y. This will be our final answer for the problem


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