# Homogeneous Equation?

## Homework Statement

$xdx+sin\frac{y}{x}(ydx-xdy) = 0$

## The Attempt at a Solution

Well, it's quite easy. But I'm quite confused if this is homogeneous or not, because of the sine function. This is my solution, assuming that this is a homogeneous equation.

let x = vy, dx = vdy + ydv; then substituting these values in the equation gives me,

v2ydy + vy2dv + y2sinvdv = 0. The variables now are separable.

$\frac{dy}{y} + \frac{dv}{v} + \frac{sinv}{v^2}dv = 0$

$ln{x} + ln{v} + \int\frac{sinv}{v^2}dv = 0$

Now, I cannot integrate the last function. I still have a lot of problems to solve, and a lot of questions to ask. I'll post others later as soon as I have tried solving them.

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HallsofIvy
Homework Helper

## Homework Statement

$xdx+sin\frac{y}{x}(ydx-xdy) = 0$

## The Attempt at a Solution

Well, it's quite easy. But I'm quite confused if this is homogeneous or not, because of the sine function.
It is sin of y/x and if you change both x and y to vx and vy, vy/vx= y/x so the sine is not changed. Yes, this is a homogeneous equation

This is my solution, assuming that this is a homogeneous equation.

let x = vy, dx = vdy + ydv;
What? If you let x= vy, then v= x/y and your sin(y/x) becomes sin(1/v)! You want
v= y/x so that y= vx, dy= vdx+ xdv. You want an equation in v and x, not v and y.

then substituting these values in the equation gives me,

v2ydy + vy2dv + y2sinvdv = 0. The variables now are separable.

$\frac{dy}{y} + \frac{dv}{v} + \frac{sinv}{v^2}dv = 0$

$ln{x} + ln{v} + \int\frac{sinv}{v^2}dv = 0$

Now, I cannot integrate the last function. I still have a lot of problems to solve, and a lot of questions to ask. I'll post others later as soon as I have tried solving them.

## Homework Statement

$xdx+sin\frac{y}{x}(ydx-xdy) = 0$
I think it's kind of disturbing that your differential equation (as stated) has no derivatives in it.

Can I state the solution I found for this equation?

HallsofIvy
Homework Helper
I think it's kind of disturbing that your differential equation (as stated) has no derivatives in it.

Can I state the solution I found for this equation?
It has differentials which is the same thing.
If you really need to have derivatives, the equation is the same as
$$\frac{dy}{dx}= \fac{x+ ysin(y/x)}{x}= 1+ \frac{y}{x}sin(y/x)$$
which is why the substitution v= y/x (NOT v= x/y) works.

Please give Mastur a chance to do it himself. If he makes the correction I suggested, he should be able to do it easily.

It has differentials which is the same thing.
If you really need to have derivatives, the equation is the same as
$$\frac{dy}{dx}= \fac{x+ ysin(y/x)}{x}= 1+ \frac{y}{x}sin(y/x)$$
which is why the substitution v= y/x (NOT v= x/y) works.

Please give Mastur a chance to do it himself. If he makes the correction I suggested, he should be able to do it easily.
Yes, I know, but I think it leads to confusion since usually only separate differentials when the equation is separable.

Oh, I see.

I'll try to re-solve the problem tomorrow. I'm quite tired right now because of the assigned activity for our group.

Thanks for the hints.

HallsofIvy