• Support PF! Buy your school textbooks, materials and every day products Here!

Help with solving this differential equation?

  • Thread starter lillybeans
  • Start date
  • #1
68
1

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.
 

Answers and Replies

  • #2
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,508
730
M(x,y) and N(x,y) are both homogeneous of degree 4. So y = ux will make a separable equation of it.
 
  • #3
68
1
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
 
  • #4
LCKurtz
Science Advisor
Homework Helper
Insights Author
Gold Member
9,508
730
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.
 
Last edited:
  • #5
68
1
[solved]
 
Last edited:
  • #6
68
1
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! :)
 

Related Threads for: Help with solving this differential equation?

Replies
2
Views
2K
  • Last Post
Replies
2
Views
937
  • Last Post
Replies
6
Views
3K
  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
7
Views
2K
Replies
2
Views
936
Replies
2
Views
1K
Replies
12
Views
581
Top