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

Answer

Related Courses

Calculus 2 / BC

NCERT Class 12 Part 2 - Math

Chapter 9

Differential Equations

Section 5

Methods of Solving First order, First Degree Differential Equations

Discussion

You must be signed in to discuss.

Top Calculus 2 / BC Educators

Watch More Solved Questions in Chapter 9

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Video Transcript

we have this differential equation which we will now prove that it is homogeneous. And then we will solve it. So here I have outlined a few steps that we could follow to prove that differential equation is homogeneous. So in the first step we write down the given differential equation as do a body X equal to f of X. Com away. And then we find four of lambda X. Come along the way by replacing, expel every X and Y well underway. And finally we check out if four of lambda X comma lambda Y Is equal to four. It's come away. We then say that the differential equation is homogeneous. So let me do the first step, since I need to have dvds on the one of the site, preferably on website, I'm going to transfer this complete turn to the side. And then I have extreme softly way by dx on the website. So this will be white minus of X. Times of sign off boy boy X. On the right side. Now we can do it both sides by X. So when I do that this X will get removed from here and it will get divided like this. So now we have returned this uh differential equations in the O. B. D. X equal to four X. Come away. So basically this right side of expressions, the president's the F of X. Come away. You know, find a 4th lambrecks. It is let's do this against him. You have to find a 400 x. Kamala Mala. Well for that we replace uh this F of X comma Y. All the excess lambrecks and all the wisest away. So let me do that. I can replace this Y. S family away minus X. Will be under X. Times of sign off number. Why by lambda, X. All over this X will be replaced as lambda X. We can see that we can factor it and then cancel the lambda. So we can divide this lambda lambda with this lambda. And also this lambda and Islam will also get canceled. Which means I'll be getting the expression y minus X. Times or sign up Y by X over X. So this is equal and two F of X com away as you can observe here, which means this F of X. Common camera white Is equal to the 4th X come away. So this confirms that the differential equation is homogeneous. We know see how to solve this differential equation. We're going to ah understand few steps. So let me load the steps here. So basically we'll be following these steps that is in the first step we have to write this. Uh We have to make the substitution Y equal to V. X. And then we find a dy dx uh for from this one that we will get in terms of X. And we uh and then we substitute this DVD X into the original DVD ex uh listen to the original differential equation so that we'll be getting an equation in terms of we and X. And then we apply this variable separable method to solve forward? So first let me right on this DVD X which we have just found out. So this is our DVD X. I'm going to write down here. So we have on the right side, Y minus X. Times of signed by by X. So left side we have do you have I D X. So can I donate a bit clearly? This is the way by dx. There is a call to Yeah, y minus X. Times of sine wave. It's so it is why minus X. Times of sign off. Bye bye X. All over X. So this or this is what we have for diabetics. It's called this equation one. We don't make the substitution. Why equal two weeks. And we debate this that is do you embody X. On the right side will be getting we plus X times of D V by the X. When you play the product tool of differentiation to differentiate weeks. So we can consider this situation too. Then we replace this dvds from equation one as system. So let me do that. So on the website I'll be having B plus extremes of D V by dx is equal to on the right side since I'm making this operation, why called Vieques, I replaced Y S V x minus off. How extensive sign of webex. So extreme self signed off. So from this in place uh this is we can solve for why that is we call to y by X. So basically why buy extra presents? Right so I can replace this way by X here as we which I'm going to do now. So therefore I'll be getting extra himself signed me do that boy X let me get this simplified. I can see that I can do it both numerator and denominator by X. This X and this X will be canceled this one and this one we can sir so we'll have the new uh right side as we minus sign me and they said we have this we plus X times of devi by dx. Now we subtract away from boats is so therefore these two weeks will be cancelled then we simplify this is extremes of devi by dx is equal to negative sign me. Okay now let me separate the variables that is can write on this as we divided by sine we is equal to negative of D X by X. So now we have successfully separated the variables that is all white victims on the website and all extremes under they've seen. So this is one way sign me which is uh cause he can't we can write on this as because he can weep times of tv is a cartoon minus of D X by X. Now we can integrate both says so you must be knowing this formula for the integration of course I can't let's say Christie to integration of Corsican digital digital. This is equal to log Corsican Theta minus carpet all Okay, so therefore when we integrate this Corsican to be we'll be getting a log off because we can't we minus cart we this is under the absolute value and this will be having minus log see log X plus this integration concert, I'm going to put this as a proxy. So let's get this simplified. In fact we can simplify this right side as a log off. See by excusing the property of logarithms. So let me write on this as log off. Sieber Ex this is a logo. See by X. Yeah. Now we can remove the algorithm from both sides when we do that we'll be getting a cozy can't of v minus court of we is equal to Sieber ex we know do the back substitution for weight because I'm sorry before we because we made the substitution Y equal to vieques which implies we equal to well my ex we substitute for we know so therefore this will become cozy. Can't off. Why buy X minus caught off. Bye bye X. This is a call to see by X. You know, convert this website. Everything in terms of uh sign and costs by using the normal trick identities. Remember that cozy can't of Tito Is equal to one by Sine Theta. Also caught it off equal to cost it or over 70 talk. So therefore this corsican weber X will be replaced as one by sign off Y by X. When is this court? Webex can replace? This is cause of web X over. Sign off by by X. And this right side remains the same. That is C by X. Now we see that we have this common denominator sign way back so we can combine these two factions and we can reduce this is one minus cause of why buy eggs over. Sign off way by X. And this is a call to see by X. So let me multiply both sides by sign off webex. So when I do that I'll be getting one minus cause of Y by X. Is equal to see time. Self sign off, Y by X. Do it by X. But I'm going to multiply by X so therefore I can read on this as X time. So Extremes of 1- Course of Weber It's is equal to three times of sign up my bags. So this is the solution after Cuban differential equation

We have video lessons for 89.33% of the questions in this textbook