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ODE, seperation of variable

  1. Sep 23, 2010 #1


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    1. The problem statement, all variables and given/known data
    Verify that the following ODE can be reduced to an ODE of separable variables.
    [tex]\frac{dy}{dx} =f(ax+by+c)[/tex] where a, b and c are constants.

    2. The attempt at a solution
    I think I must show that there exist functions g and h such that [tex]g(y)dy=h(x)dx[/tex].
    I have that [tex]dy=f(ax+by+c) dx[/tex]. I was at a loss. So I talked to a friend and he told me to write [tex]u=ax+by+c[/tex].
    So I get [tex]dy=f(u)dx \Rightarrow y= \int f(u)dx=\frac{u-ax-c}{b}[/tex], [tex]y'=\frac{u'-a}{b}[/tex], [tex]y''=u''[/tex]. I want to write [tex]f(u)[/tex] as [tex]\phi _1 (x) \phi _2 (y)[/tex] but I'm totally stuck.
    I'd love a tip.
    Thank you.
  2. jcsd
  3. Sep 23, 2010 #2


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    You have figured out that y' = (u'-a)/b and if you plug that in for y' you get

    (u'-a)/b = f(u)

    Write u' as du/dx and, remembering that a and b are constants, see if you can't get the u terms and the x terms on opposite sides.
  4. Sep 23, 2010 #3
    ya lckurtz sums it up

    also try looking here:http://www.tutorvista.com/math/separable-differential-equation" [Broken]
    Last edited by a moderator: May 4, 2017
  5. Sep 27, 2010 #4


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    Thank you guys, I got it.
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