Help with solving this differential equation?

[lillybeans]

Homework Statement

(3y⁴-11x²y²-28x⁴)dx-(4xy³)dy=0

Homework Equations

My=Nx if equation is exact, if not, I can make it exact (I hope) by finding an integrating factor. The problem is I can't get the integrating factor to be a function of x or y only.

The Attempt at a Solution

Let stuff in front of dx=M
Let stuff in front of dy=N

My=12y³-22x²y
Nx=-4y³
My-Nx=16y³-22x²y

First try (My-Nx)/N=(16y³-22x²y)/4xy³

Not a function of x only.

Then (My-Nx)/-M=16y³-22x²y/-(3y⁴-11x²y²)

Not a function of y only.

Am I using the wrong method to solve this question? I don't see any other way to go about it rather than turning this into an exact equation. Why can't I find an integrating factor that has a pure x or y expression?

Thanks.

 

[LCKurtz]

M(x,y) and N(x,y) are both homogeneous of degree 4. So y = ux will make a separable equation of it.

 

[lillybeans]

Sorry, I haven't quite learned that yet. Do you mind elaborating a little? Is that another technique for solving differential equations? Something called substitution, maybe?

Thank you

 

[LCKurtz]

A function $f(x,y)$ is homogeneous of degree $n$ if $f(\lambda x,\lambda y) =\lambda^nf(x,y)$. What $n$ works for your $M$ and $N$?

Try the substitution $y = ux,\quad dy =xdu + udx$.

[Edit, added later]: I see I forgot to tell what this type of DE is called. If is called a first order homogeneous equation. Unfortunately, this is not the same meaning of the term as when applied to a linear DE with 0 on the right side, where we refer to homogeneous vs. non-homogeneous equations. In the setting of your problem, it refers to the $M$ and $N$ being homogeneous in the above sense.

 

[lillybeans]

[solved]

 

[lillybeans]

Nevermind I made a mistake, that's why I thought I couldn't turn the last one into an exact one.

Thank you once again! :)