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Partial fractions (?) to solve first order DE

  1. Aug 17, 2012 #1
    hello world,

    I've been doing some summertime training to brush up my math skills and have been struggling with this


    In fact this is the simplified version of a Bernouilli equation. I know that it is separable, I'm just struggling with the integration with respect to "y".

    If anybody could help that'd be wonderful. Thanks a lot, have a nice day!

  2. jcsd
  3. Aug 17, 2012 #2


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    Your equation is
    [tex]\frac{dy}{dx}= \left(e^{-y}+ Ce^{-2y}\right)^{1/2}[/tex]
    which can be written as
    [tex]\frac{dy}{\left(e^{-y}+ Ce^{-2y}\right)^{1/2}}= dx[/tex]

    I would let [itex]u= e^{-y}[/itex] so that [itex]du= -e^{-y}dy[/itex] and [itex]du/u= -dy[/itex] In terms of u, the equation becomes
    [tex]\frac{du}{u\left(u+ Cu^2\right)^{1/2}}= dt[/tex]
  4. Aug 17, 2012 #3

    Thank you for you response.

    I tried your method and i've run up against a similar problem as before, when i tried partial fractions. I can't seem to solve correctly for the coefficients.

    A/u +B/(u+C*u^2)^(1/2)=1

    A*(u+C*u^2)^(1/2) +Bu=1

    choosing u=(-1/C) --> B=-C

    however, I get stuck looking A, the only way to make the B drop is to set u=0, which is not coherent.

    do you have a hint for me? thanks.

  5. Aug 17, 2012 #4
    ps ( how do you enter the equations so nicely?)
  6. Aug 18, 2012 #5


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    I don't believe you can use partial fractions when there are surds involved.
    Try making a change of variable to get the surd in the form √(u2+B) (where B may be negative in this case), then look for a trig or hyperbolic trig substitution to make the surd collapse.
    To make your posts neater, click on "Go Advanced". That brings up a palette on the right from which you pick various symbols, and a toolbar above which makes e.g. superscript and subscript easy. To make it really pretty, click on the Ʃ symbol at the end of the toolbar. This brings up a Latex palette. You'll need to play around with that a bit to get the hang of it. If using either of these, remember to click Preview Post and check what it's going to look like before submitting.
  7. Aug 18, 2012 #6


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    Dearly Missed

    At some point, after completing the square and having eliminated the surd through haruspex's advice, you probably will need to make another substitution of te form v=Tan(u/2), or v=Tanh(u/2).
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