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Integrating factor ode

  • Thread starter manenbu
  • Start date
  • #1
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Homework Statement



Solve:
[tex](1-\frac{x}{y})dx + (2xy + \frac{x}{y} + \frac{x^2}{y^2})dy = 0[/tex]

The Attempt at a Solution



No idea what strategy to use here. Tried using an integrating factor, but no success. A lot of x/y in here makes me think I need to use a substitution, but there's also "xy" in here which doesn't help me. What should I do?
 

Answers and Replies

  • #2


How did you try using the integrating factor? Did you try multiplying by "x^m*y^n" and then solving the integers "m" and "n"?
 
  • #3
103
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I tried using the formulas:

[tex]
F(x) = \frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N}
[/tex]
And then:
[tex]\mu(x) = e^{\int F(x) dx}[/tex]

(there's also a similar one for y, with signs reversed and M instead of N in the denominator).

Had trouble in the F(x) part - couldn't get a function of x only (or y, for that matter).
 
Last edited:
  • #4
103
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Ok! Solved it. Missed a little thing on my side.

Thanks anyway! :)
 
  • #5
gabbagabbahey
Homework Helper
Gold Member
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The substitution [itex]u(y)=\frac{x}{y}[/itex] works out nicely.
 
  • #6
103
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[tex]C = \ln{|x|} + \ln{|y|} - \frac{x}{y} + y^2[/tex]

That's the solution, in 2 different strategies.
One using the integrating factor, second using your substitution. Thanks for pointing it out. :)
 

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