In each of the Exercises 1 to 10 , show that the given differential equation is homogeneous and solve each of them.
$x^{2} \frac{d y}{d x}=x^{2}-2 y^{2}+x y$

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Calculus 2 / BC

NCERT Class 12 Part 2 - Math

Chapter 9

Differential Equations

Section 5

Methods of Solving First order, First Degree Differential Equations

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Video Transcript

will not prove that this differential equation is homogeneous. And then we will solve it So to prove that given differential equation is homogeneous. We follow these three steps that is in step one. We write down the equation in the form of D um B D X equal to f of X. Com awake and it is against him. We find F of lambda X. Come on camera. Why replacing the X by lambda X and Y well underway. And in the third step we check if 1/4 lambda X come on camera. White is equal to a four X. Come away. We then say that the given differential equation is homogeneous. So let's go ahead and check this observed that on the website of this equation we already have this dvds but we also have this factor escort. So you just have to divide both sides X squared. So that we will get you every day X on the right website. So which I'm doing now. So I can read on the right side as uh X squared minus two Y squared plus ex wife. All over X squared. So now we can say that this is in the form of Dvds. So let me call the right side as a four X com away. So the test I'm going to read on this right side expression as a four X comma Y. Because it's a function of both the X and Y. This is too I squared plus xy over X squared. So we have done this first step and in the next step we had to find the F of lambda X comma on broadway by replacing the X by lambda X and Y by lambda. We will do it now. So the first term in the numerator is X squared. When we replace it with the lambda X. That will become lambda square X squared. And the next term is -2 waistcoat. If you replace why with the land away it will become camera square, why square plus this X will be lambda X. And that's why we'll be on our way. So we get like this in the numerator and denominator we just have X squared. So therefore if you replace it with the land rights, it will become, I'm not scared export. So we can simplify this by factoring landlords called in both numerator and the denominator. So when we do that we'll be getting lambda squared and x squared minus two Y squared plus this is basically I'm rascal that is the third time. So we'll be left with X Y term over time. That's good X squared. So I can cancel this lambda square like this. So therefore it now looks like we got this F of X com away. So if you observe this, this term is completely Your four x come away. So we were able to see that this is equal and two the fourth that is 1/4 lambda X comma, Y is in fact equal to four X comma, Y. Which means we are able to show that it is a homo genius. Now we are going to solve the differential equation. I'm actually loading the steps uh We could follow to solve the differential equation, which we have just proved that it is a homogeneous. So these are the few steps. So the first step we basically Make the substitution why he called two weeks. And then we immediately follow. We immediately find the D O. B D X right down in terms of X. Henry. And we substitute the dvds which we have found in the step to into the DVD X uh uh equation and uh we then uh apply the variable separable method and solve it. So let's go ahead and do it. I'm going to ride on this initial equation. Now that is uh meant to write on this equation. This is our original differential equation. So it's just the way by dx is equal to enumerated. We have x squared minus two Y squared plus xy X squared minus voice squared. I'm sorry, this -2 ways square plus. Xy over X. Quipped. Okay, so let's call the situation number one. Then we make the substitution by equal two weeks. And then we differentiate this immediately. When we do this we'll be getting D um B D X equal to we plus X times. So DVB dX we have to play the product to to differentiate the VX. So let's call this a situation to observed that ah these websites are equal or otherwise we can replace this dy of this equation one as uh this this terms. So let me do that. So on the left side I'll be having replaced extremes of D V by dx is equal to X squared minus two times Y squared. I'm going to substitute Y S V X. So therefore this will be re squared X squared plus X. Times of we have white as an equation one. So therefore this will become VX all over X squared. So let me factor this X squared from the numerator. When I do that I'll be getting 1 -2 weeks squared plus read all over X squared. So this X squared and X squared will get cancelled. Let me relate this equation. So this is we plus X times of devi by dx is equal to 1 -2 We squared plus we we can cancel this week. That is if you subtract we from both sides this will become cancelled. Now let me relate this situation X D v by dx Called to 1 -2 We Scott. So now we have to separate the variables that is all the victims and the T V should should come on the website which means we have to divide both sides by this factor one. Minister B squared. And when I do that I'll be getting one, do it by 1 -2. We squared times of devi this is equal to I multiply by dx and divide by X. So big having the X over X in the right side. Now we have separated the variables, that is all the victims and the day we are on the website and all the extremes are on the right side so we can integrate both sides like this. We put this integration symbol. Um so to integrate this, we are going to make some adjustments. Uh I'm going to write down this factor, this is one x 1 -2 we squared. So this can relate as if a factor too from the denominator this will be basically had to divide by two minus softly square. So this is what I have been a factor there too and then I'm going to relate this uh half it is at this half, I can really this as one way route to all square minus b squared. So I read it like this so which uh so when I do that I can replace this equation. That is an integration equation as uh like this that is, we have Davey in the numerator and the denominator. We have two times. So one by route to square minus we square integration and just say I can integrate one by ex uh so we'll be getting log off absolute value of X plus C. Okay now we can trace the formula to integrate this at least that this is no form of one by a squared minus X squared the X. We can use this integration formula and the integration formula, it says one way to a logo, A plus X Divided by a -6 and absolute well so I'm going to use this integration formula and when they do that this left side I'll be having one way to claims soft one by two years. So I put one by two. This is there's one Beirut to is basically the eight because uh we can see that it is in the form of a square minutes minus b squared so therefore in in place of a I can right right one way route to. So I have completed this spot and then I have logged off E plus six is basically we can write down here one by route so therefore I write down one by route two plus instead of facts we have to put we because it's functioning we Do it by one by route to might not be like this. So this will be equal to we have a log X plus C on the right side. So let me rewrite this equation by simplifying. This is so this will go route two times. So basically we have one way to route to as the multiplication factor time. So log off. This can be written as one plus route to V divided by route to and soon nearly one minus route to. We divided by Ruutu absolute value. So decide we have a log x less C. Now let's get this simplified uh what he said we have this denominator route to for the numerator fraction and similarly for a route to for the fraction in the denominator which means we can cancel them both. So the next step we can read on this as one x 2 route to plain soft log off one plus I'm going to replace for we know that we made a substitution. Why called we uh we're called to Vieques Actually we have made UAE called V X in the beginning which means y equal to I'm sorry we equal to Y by X. So we back substitute for a week. I'm going to subscribe to weber X Similarly this will be one minus through two times. So bye bye X absolute value. So this is equal to log x. Let's see let's get this simplified. So this I will be having one x two route to logo. And when you simplify this fraction in the new mandatory has placed in the denominator we'll be having express route to times of hawaii Divided by X- Route two times. So white. This is absolute value. This is the call to log X plus C. Okay so we have to put this absolute value. So therefore the system solution to the given differential equations

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