First order ODE, The Homogeneous Method.

  • Thread starter knowlewj01
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  • #1
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Homework Statement



[tex]\frac{1}{xy}[/tex] [tex]\frac{dy}{dx}[/tex] = [tex]\frac{1}{(x^2 + 3y^2)}[/tex]


Homework Equations



used the substitutions:

v = [tex]\frac{x}{y}[/tex] ,and
[tex]\frac{dy}{dx}[/tex] = [tex]v + x[/tex] [tex]\frac{dv}{dx}[/tex]

The Attempt at a Solution



took out a factor of xy on the denominator of the term on the right hand side and multiplied through by xy, made the substitutions and at the moment I have something that looks like this:

[tex]\frac{1}{v + 3/v}[/tex] = v + x [tex]\frac{dv}{dx}[/tex]

as it stands I cant see a way to do this by separation of variables, is it solvable by integrating factor? anyone got any ideas?
 

Answers and Replies

  • #2
35,125
6,872
You used the substitution v = x/y, but your work in calculating dy/dx looks like you started with y = vx, which is not equivalent to v = x/y.
 
  • #3
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ah thanks, i'll give it another go.
 
  • #4
35,125
6,872
I was able to get the equation separated with the substitution v = y/x.
 
  • #5
110
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I have the substitutions:

v = [tex]\frac{y}{x}[/tex]

[tex]\frac{dy}{dx}[/tex] = v + x [tex]\frac{dv}{dx}[/tex]

so now my equation looks a bit like this:

[tex]\frac{1}{1/v + 3v}[/tex] = v + x [tex]\frac{dv}{dx}[/tex]

I played with it a bit and still cant see how to seperate it, can anyone point me in the right direction?

thanks
 
  • #6
35,125
6,872
What you have is fine, but needs cleaning up using ordinary algebra techniques.

  1. Rewrite 1/(1/v + 3v)) as a rational expression - one numerator and one denominator, with no fractions in the top or bottom.
  2. Subtract v from both sides, so that you have x*dv/dx on one side, and a rational expression on the other.
  3. Divide both sides by x. You should now be able to move dv or dx so that everything in v or dv is on one side, and everything in x or dx is on the other. The equation is now separated, and you can integrate to find the solution.
 

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