In each of the Exercises 1 to 10 , show that the given differential equation is homogeneous and solve each of them.
$(x-y) d y-(x+y) d x=0$

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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

Yeah, here we have this differential equation which we will now prove that it is a homogeneous differential equation. And then we will solve it. So we'll be following these three steps to show that it is a homogeneous differential equation. So in the first step we had to write down the given differential equation. And everybody Mexico to a four x come away from. So for that we have to first to transport this term to this right to this right side. So when I do that, I'll be getting X minus boy. Times of the way is equal to X plus Y time. Soft dx. Now I divide both sides by this top. That is the X minus way. So that I'll be getting diva is equal to X plus. Why times of deals over X minus white? Once again, I divide both sides by this time so that we can get dvds on this right side. I'm sorry on the website. So when I do that, this will get removed from the right side. So here we'll be having the X. So now this is in the form of DVD X. So therefore this right side represent the F four, F X. Come away. So you can write down a four X comma Y is equal to express boy over X minus white. We're done with this first step. And in the next step, we have to find the F of lambda X. Kamel AMR away, which we will do now. So find a fault lambda X. Come on lambda. Well, which means we have to replace the X by lambda X and Y by the way. Uh in the F of X com away. So when we do that, we'll be getting lambda X plus uh that's why we'll get replaced mr land away. And similarly the denominator will have lumber X minus Lamar away. Now I can simplify this by factoring out Lambro from both numerator and denominator. So I do like this for the denominator factor, lambda hybrid getting x minus one, which means they can cancel the Lando's. So we can we'll see that it gets removed. So finally we are getting this form so which means this is equal and to a four X com away. As we can see from here, this X plus Y over x minus way. It is something but it's comma before fix come away. So you could see that from this. We see that F. F. Lambda X comma. Lambda Y is basically equal to f off X comma Y. Which means the given differential equation is a homogeneous differential equation. Well, no go ahead and solve the differential equation. Let me write this equation. We have the way by the X is equal to X plus Y Over X- where it's called. The situation is # one. We'll be following few steps to solve the differential equation. So let me put it over here. So these are the steps that we are going to follow to solve the differential equation. So first we make the situation why called a V. X. And then find everybody X. In terms of X. And we and then he quit uh DVD X. Uh I mean substitute the dvds into the original question. So convert everything in terms of excess for us. We and then you think the variable separable method will solve it. So let's do the steps. So we first make the substitution why equal two weeks? And then we differentiate this equation with respect grades. We'll be getting devoured by the Mexico too. Apply the product rule to differentiate this week's when we do that, we'll be getting we plus X times of DvB T X. Say this execution number two. Now comparing both equation one and two, we see that uh their websites are equal. So therefore the right sites will be equal, which means I'm going to I don't know uh this one we plus X. D V by dx is equal to X plus. Why remember that? We have already replaced? Uh We are already making the substation. Why could be so I'm going to replace Y S V X. So I don't like this. We X plus V X Divided by X -Y. Call two weeks now a factor X from uh numerator and denominator. I'll be getting one plus B. They were that boy X into 1- Week. We can cancel the excess. I mean I don't below this. So how we plus X. Times of devi by dx Stick all to one plus we They would advise 1- week. So now let's uh supplied to be from both sides. So we have X. D V by dx as a call to one plus we buy one minus three minus week. So this is equal to one plus we minus three times up one minus three over 1-. Well let's get the arches simplified. So we'll be having one plus we we do the distribution here minus three times of oneness minus one and minus three times minus. We. So forced to be square divided by one minus week. This plus we and minus will get cancelled. So basically we get one plus B squared over one minus we and on the left side we have x dvds. So let me I don't work here. So we have X D V by dx. This is equal to one plus we squared by one minus we. Okay, so the next step we have to separate the variables. That is all the victims has to come on the website and all the extremes we have to take it to the right side. So for that I'm going to multiply by the reciprocal of the arches. Okay, so when I do that I'll be getting one minus. We over one plus B squared times of levi and then I take this x times of the X reciprocal, I take it to the right side which means I'll be getting uh D X by X On the right side. So now this is enough form of differential equation with we and X. That is all the victims on the website and all the extremes on the right side we are separated the variables we can now go ahead and integrate. So that is actually part of the variable separable method. So integrate both sex. So this uh website I'm going to read on this uh to integral. That is one plus we square baby minus we over one plus way square integration devi on the right side. If we integrate one by X will be getting log X plus the constant of integration is C. So this in place to integrate this foster to this is using the standard integration formula. This is nothing better than in well Sophie and for this we can replace the methadone substitution. So let's do that now. So it will be uh I'm going to make the substitution. Um one plus we square let me do it here. That this one plus B squared myself to T. Then to be I take differentials on both sides to be D. V. Is equal to D. T. So therefore we D. V is equal to assault for VDV. This is equal to D. T by two. So that I can replace this VD because I have it in the numerator. So in place of VDV if I substitute DT by two this will be D two by two divided by one plus B squared S T. Okay, so we get like this this is equal to on the right side. We have log X plus C. So this is for substitution. So just in place turn inward so I can replace this, B S, Y by X. Because we made the situation why equal two weeks and solving for we we get we equal to way back so we can back substitute for me and minus of this is D. T by two is half. And then we'll be getting DT by B. Which means we have to integrate one over T. So integration of one over T is lofty. I can replace it here. This is a log off Which is 30 is nothing but one plus we square this is it called to log x. Let's see. So it's kate is simplified. We have to make sure that all the visa replaced tests Y by X. So we have a ton inverse of Y by x minus. How far log off. Let's simplify this over here. Uh, we know that we called Wiberg. So therefore this will be one plus Y squared by X squared. So when we simplify this, this will be X squared plus y squared. Well what X squared 2nd. I don't uh log off x squared logo for X squared plus Y squared. Boy, X squared is equal to on the right side. We have the symptoms. That is log X plus the control of integration. And the next step we can keep this turning values of Y by X on the website. And I transport this the storm to the right side. We're adding the same term which means I'll be getting this log X. I went to write on this log X. This excess uh X squared raised to the power of one way to so that when I do that here many substitute access X squared raised to the power of one way to. And using the property of algorithm we can bring this term over here. That is one way to we'll get multiplied with log so therefore I can read on this bus. Uh half of log of X squared plus this time the term that we have transport which means their second. I don't answer to all go X squared plus Y squared over X squared plus the constant of integration, seek and uh the next step I still have this turning words of antibiotics on the website. So now where can combine these two logarithms? Because I have the same uh multiplication factor one by two. So let me factor this one by two which means a factor. This will be getting logo X squared plus log off this term X squared plus Y squared by X squared. Which means these two will get multiplied is in the property of logarithms. So we'll be getting X squared dreams of X squared plus Y squared by X squared. Inside the log rhythm, which means these two will get cancelled, and I'll be left only with logo, X squared plus Y squared. Okay. Prestes integration Constancy. So therefore this is the solution to the Cuban differential equation.

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