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Point) Solve the equation(28xy+ 35x4y - 2y6 lOxyf )dx (Tx" 12xy5 )dy = 0leaving your answer in implicit form: The solution iswhere C is constant of integration...

Question

Point) Solve the equation(28xy+ 35x4y - 2y6 lOxyf )dx (Tx" 12xy5 )dy = 0leaving your answer in implicit form: The solution iswhere C is constant of integration Note: Enter an answer that is valid function of x and y defined on the whole plane_

point) Solve the equation (28xy+ 35x4y - 2y6 lOxyf )dx (Tx" 12xy5 )dy = 0 leaving your answer in implicit form: The solution is where C is constant of integration Note: Enter an answer that is valid function of x and y defined on the whole plane_



Answers

Find the particular solution to the differential equation, given the general solution and an initial condition. ( $C$ is the constant of integration.) $y=\sqrt{2 t+C} ;$ the solution curve passes through (1,3)

So in the given question we are to find the particular solution to a differential equation. Both general solution is of Isaac was to its graduate of duty plus C. And the solution girl is passing through the point one commentary. Right? So the approach here, this whole discussion is to find a particular solution, we have to find the value of the constant C. Right? So it is given that the girl is solution covers passing through the .1 commentary. It means that the particular solution would be satisfying. Like the condition that when The use equals to one why is equal to three. So this value we can substitute here, you will get three Sequels to its credit of two into one plus C. Right? So from here we get the value of CS seven. This value of the weekend substitute in the general solution. So we will get Y is equal to the square root of 2-plus 7. This is the particular solution to the differential equation. Yeah.

This question asked us to find the solution of the differential equation that satisfies the given initial condition we have D y over D backs is axe sign acts divide by. Why Now we know we we have to multiply both sides by why, in order to get all the y stuff or y terms on the left hand side, and then we know we can multiply both sides by D. X in order to get the ex terms on the right hand side like this. Not integrate. We know why becomes why squared over to us. We increase the exponent bar one divide by the new exponents and on the right hand side, we have negative acts Coastline X, plus sign, X pussy or constant. Now remember, we want this just in terms of singular Why? Which means we have to multiply both sides by two and then take the square root to get rid of the square root and the 1/2 as wth e coefficient we end up with Why is poster minus square root of negative two acts co sign acts plus to sign experts See, now we know they've given us an initial value of y of zeros. Negative. Want we can use this plug in wise negative one x zero we're solving for C. Don't forget. Once we saw first, see, we're gonna be plugging it back into the original equation. This simply means that seat is one. Because we know we're looking at simply negative one is negative square root of sea. We disregard the positive solution because that wouldn't logically sense. It only works if it's negative. Squirt of C two c is one now. Like I said, we're plugging the spot into the equation. We'd figured out why is negative square root of negative two acts co sign X plus to sign acts. The only difference is we're just adding our seed, which is plus one.

Okay this is the differential equation that were given and uh so I'm going to let U. Equal to X. Plus one. So then D. U. Is going to be to you D. X. And therefore D. X. Is going to be one half D. You. So D Y. Is going to be U. To the fourth power times dx. Which is one half D. You therefore why is going to be U to the fifth power over five times one half mm Plus C. Now I said that you was two X plus one so it's going to be two X plus one to the fifth power over 10 plus C. Now we know that Y equals six. When X equals 16 equals two plus one to the fifth. Power over 10 plus. See six equals three to the fifth power over 10. Oh my goodness. What is that three to the fifth power? Which is 243? Oh no X zero X zero X. Is not one X zero. So that would be zero plus one. That would be one to the fifth power over 10. So 1/10. Okay. Six minus 1/10. That would be 59 10th. So where were we right here, Two X plus one to the fifth Power over 10 plus 59 over 10. That's it, Yeah.

Yeah. Do you Why over dx is the square root of X over the square root of why? So I'm going to write that as X. To the one half power over Why? To the one half power. Then I'm going to get all the wise on the same side and all the axes on the same side. I'm going to integrate both sides. Um Why to the three has power over three halves? Uh Two thirds X. Yes. Power. And I'm going to write plus D. Because then I'm going to just divide everything by two thirds and right see okay now why is going to be X to the three halves plus C. To the two thirds power? Okay we know that why equals four when X equals one. So four is going to be one to the three halves power plus C. To the two thirds power. Okay so taking uh three has power on both sides. I get four to the three halves. Power is going to equal one plus C. the square root of forest to um to to the third power is eight, so eight equals one plus C. So C. Is seven. And so the particular solution is X. To the three halves, plus seven to the two thirds power. Okay. Yeah.


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