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Third Order Non-Linear Homogeneous ODE

  1. Dec 5, 2011 #1
    I have derived a 3rd order non-linear ODE with its respective boundary conditions, and I was hoping to get a hint on how to find a closed form solution to it. The equation is given as:

    F''' + (1/C^2)*F*F' = 0

    Where the primes denote a derivative, and C is just a constant. Any help is highly appreciated!
     
  2. jcsd
  3. Dec 5, 2011 #2

    Mark44

    Staff: Mentor

    I think this might help. Write the equation as F''' = -(1/C2)FF'
    Now integrate both sides. That will get you a 2nd order DE.
     
  4. Dec 5, 2011 #3
    So I tried what you mentioned, and here's what I get (though I think I made a mistake somewhere):

    F''' = -(1/C^2)*F*F'
    > F'' = -(1/2/C^2)*F^2 + C1 (where C1 is a new constant)

    Now, this gives:

    F'' + (1/2/C^2)*F^2 + C1 = 0

    Assume G = F'
    F'' = G*dG/dF

    > G*dG/dF + (1/2/C^2)*F^2 + C1 = 0
    > G*dG = [-(1/2/C^2)*F^2 - C1]*dF

    Integrating both sides gives,

    G = sqrt(-(1/3/C^2)*F^3 - 2*C1*F + 2*C2) = F' = dF/dy

    Separating variables gives,

    dF/sqrt(-(1/3/C^2)*F^3 - 2*C1*F + 2*C2) = dy

    But now I have no idea how to take that integral and I'm pretty sure I made some mistakes with the constants. Any suggestions as to where I went wrong and what I can alter?
     
  5. Dec 6, 2011 #4
    You said closed-form right? First just write it as:

    [tex]\frac{dy}{dx}=\sqrt{ay^3+by+c}[/tex]

    Now, suppose I told you there is a special function called the Weierstrass P function such that if:

    [tex]\text{myxside}=\int_{\infty}^{y}\frac{dt}{\sqrt{4t^3-ct-d}}[/tex]

    then:

    [tex]y=P(\text{myxside,c,d})[/tex]

    and you just had to make some kind of showing on this thing, could you express your solution in terms of the Weierstrass P function?
     
    Last edited by a moderator: Dec 6, 2011
  6. Dec 6, 2011 #5
    Honestly, I'm not familiar with the Weierstrass P function. I have to solve this ODE and apply three boundary conditions to it. I'm not sure how the Weierstrass P function operates to be able to do that.
     
  7. Dec 6, 2011 #6
    Looking at it again, I think I found a way to solve the ODE, if the 1/C^2 constant wasn't there. Is there any way to define a new variable F that absorbs that constant value. If I can find a way to do that, then that ODE is easily solvable.
     
  8. Dec 7, 2011 #7
    Nevermind. Solved! Thanks for the help!
     
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