Can an Integrated Factor be Found in Terms of Both Variables?

[MMars91]

Homework Statement

I have been dealing with Exact Equations in my DE class, and I came around this problem.

(t^2-y^2)+(t^2-2ty)(dy/dt)=0

This is obviously not an exact eqn. So I tried using integrated factors on it and try to find this "factor" μ.
But no matter if I did it in terms of t or in terms of y, I couldn't separate it in terms of one variable.

dμ/dt=(-2t)/(t^2-2ty)

or

dμ/dy=(2t)/(t^2-y^2)

Homework Equations

Is there any way that you can find an integrated factor which it is in terms of both variables?
instead of t or y alone, both?

The Attempt at a Solution

I tried everything, and this topic is not even covered in class or in the book. I learned this on my own and I have only learn Integrated factors in terms of y or in terms of t, not both.

Help please.

 

[LCKurtz]

Homework Statement

I have been dealing with Exact Equations in my DE class, and I came around this problem.

(t^2-y^2)+(t^2-2ty)(dy/dt)=0

This is obviously not an exact eqn. So I tried using integrated factors on it and try to find this "factor" μ.
But no matter if I did it in terms of t or in terms of y, I couldn't separate it in terms of one variable.

dμ/dt=(-2t)/(t^2-2ty)

or

dμ/dy=(2t)/(t^2-y^2)

Homework Equations

Is there any way that you can find an integrated factor which it is in terms of both variables?
instead of t or y alone, both?

The Attempt at a Solution

I tried everything, and this topic is not even covered in class or in the book. I learned this on my own and I have only learn Integrated factors in terms of y or in terms of t, not both.

Help please.

A function $M(x,y)$ is said to be homogeneous of order n if $M(\lambda x,\lambda y)= \lambda^nM(x,y)$. If $M(x,y)$ and $N(x,y)$ are both homogeneous of degree n, then the differential equation $M(x,y)dx + N(x,y)dy = 0$ can be converted to a separable DE with the substitution $y=ux$.

That applies to your question. Look at http://www.cliffsnotes.com/study_guide/First-Order-Homogeneous-Equations.topicArticleId-19736,articleId-19713.html for a discussion of this type of equation.

 

[MMars91]

Thanks, that just lighted a bulb in my head.