In each of the Exercises 1 to 10 , show that the given differential equation is homogeneous and solve each of them.
$1 e^{\frac{x}{y}} d x e^{\frac{x}{y}} 1 \frac{x}{y} d y \quad 0$

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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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Problem 1

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Problem 10

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Video Transcript

Yeah. We will not prove that this differential equation is homogeneous. And then we will solve it to prove that this is a homogeneous differential equation. We had to write down in the form of dx dy is equal to some function of X by white. So if you can write down this differential equation in this form then this is homogeneous. So let's try to write on uh as in this one, I'm going to keep this term here and transpose this term to the right side so that I can write down in the expedient form. So let me do that. This is one plus E. Power X by Y. Times of dx. This is equal to this will be negative, which means uh This will be explained why I can write on this, explain why will become past two and the one will become negative. So like I write like this the power explained why. And then I have this feeling. No I had to do it by this time. A serious by the day waiter. So that I will get in terms of the expert away. So let me do it over here. So I bring this early way for divide. Uh I'll get this uh He explained you were here and then I had to do it by this time one press report, explain why. So let me do that. Just a stupid boy. one plus e. Power X by Y. So as you can absorb all the terms has X by Y form, which means this is basically nothing but some function of X by Y. So we can see that we have returned this in the form of the expert diver is equal to some function of X. Y. Y. Which means this given differential equation is homogeneous. We will not solve this differential equation for that. We have to make this substitution X equal to be way and then we differentiate both sides with respect to why? So if we differentiate X with respect to Y. We'll be getting dx by DY. On this side. We have to use the product rule of differentiation to differentiate with respect to white. So we will get me time. So differentiation of Y with respect to Y. Is one plus Y. Times of differentiation of E. With respect to why that is study we've anyway so we can replace this the expert the way as this expression. Let me do that. So on the left side I'll be having replace white times of D. V by Dy is equal to I'm also going to use this substitution and find X by Y. From this. X by Y. Is basically we if you divide both sides by wai brigade. This form which means in this expression wherever I have this X by Y. I can replace it with which I'm going to do now. So this will be we minus one time soft. The numerator, I'll be having re -1 time. So he raced to the poor ofi divided by the denominator. We have one plus epo ex explain why? Which means it will become one plus E. Power explain why replace it last week. So this is what we get now. Let's subtract away from both sides so I can remove this week And for this as uh negatively over here. Let uh get this rate said simplify let me do the distribution first with people. We so it will be we times of a poor we minus of a party and then had to multiply this term with this term so that I will have the common denominator. So minus of B minus we E raised to the power coffee all over one plus E. Waste topography. So let's simplify this. We have the ipod or we and the negative people. We Then we can write on this as an a simplified form as I can factor the minus that is -1. Uh If I do that I'd be getting we plus people are weak all over one plus E. Power bi. And this side we have white time self. Do we baby? What? So let me separate the variables. So I have to bring this uh It is this so how to bring these terms to the left side. That is this complete terms to the left side. And when I do that I can multiply by its reciprocal. That is uh I read on this as one plus a party divided by we plus a power We times of baby. I keep this negative as it is. This negative will remain as it is minus. I had to multiply by the reciprocal of why? By the way which is divided by why remember that? We have this negative can put this in the same colour. So we have this minus of the U. S. B. Y. Now we have separated the variables. We can go ahead and integrated. So we put this integration symbol. Uh This right side integration is quite simple. This is integration of one. B. Y. Is log mulled wine. So you can put this as simply log off more white plus uh The integration constant that is second. I do this as a log see of Morsi. Now to integrate this we have to make a substitution. That is uh we can put this we plus people we this is in custody. Now we take our differentials on both sides. So this will be we have to differentiate with respect to be this will be one plus if power be multiplied by daily is equal to the semi. To differentiate with respect to T. We get DD. So now if you could absorb I can replace this complete terms us B. B. As I have it here. One plus april three times of D. V. Is equal to D. D. So therefore this right side. Uh I'm sorry the left side part will become DT over the speed plus the police T. So it comes out like this. Now let's keep this as it is. This is a log of absolute value of Y. That's a no go absolute radio city. Now integration of one by T. S log absolute value of T. And then I put this value of T replace the value of that is we plus he raised to the power of me. So this is equal to I can bring this uh minus log way to this side to the left side. And when I do that it will become positive. This is a log off absolute value. Why? Plus I'm sorry This is equal to love Morsi. Now let's combine both this logarithms that is that this and this using the log property that is logged place, lobbyist log off. Maybe. So therefore this implies the next step. You can I don't exist. We plus is to the power of three multiplied by Y. This is equal to knock. Absolutely idiotic. But you can remove the algorithm from both sides which means we will get we plus E raised to the power of we time. So why is equal to see now we'll do the back substitution for we remember that we made the substitution X equal to B. Y. So from that from this we can find value for me that is we call two X by white. So we do this backs electrician for me. So we're going to replace all the visas expire right. Therefore the silver X by Y. Les E raised to the power of X by Y. Times. Of why he called to see, let's get this simplified. This will be express why terms of E raised to the power of X by Y. We had this way in the denominator. And that's why he called to see if I can cancel these two ways. And finally, I'm getting express. Y. E raised to the power of X by Y. Is he called to see? So this is the solution for the given differential equation.

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