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Differential equation I can't solve

  1. Sep 1, 2007 #1
    I was solving my own-invented physical problem, and obtained a differential equation
    [tex]\frac{dx}{dt} = k \sqrt{t-x^2}[/tex]
    with k positive constant. I wish to solve it. I'm unable perform separation of variables here, and power-series method works poorly, too. I conclude that this equation does not have any closed-form or familiar series solution, so the best thing that I could do is use numerical methods [of an Excel type, but Gnumeric is my choice :)]. It turned out that with k=1, when t=1, x=0,6, exacty. This gives a clue that there might exist some sort of a solution (or it might not be any kind of a clue, I'm just being silly). How do I find it? How do I solve this equation?
  2. jcsd
  3. Sep 2, 2007 #2


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    There's no exact solution. I'm assuming you're using a boundary condition of x=0 at t=0. Then, Mathematica's numerical routine NDSolve gives x=0.60635 at t=1.

    You can solve it approximately by Taylor series for small t and large t. At small t,
    [tex] x = {{\textstyle{2\over3}}t^{3/2}\left(1-{\textstyle{2\over21}}t^2
    +{\textstyle{10\over2079}}t^4 + \ldots\right) [/tex]
    At large t,
    [tex] x = t^{1/2}\left(1-{\textstyle{1\over8}}t^{-2}
    -{\textstyle{13\over128}}t^{-4} + \ldots\right) [/tex]
    These cross at t=2.354, where they differ from the numerical solution by a worst-case 1%.

    This is for k=1. But if [tex]x=f(t)[/tex] solves the equation for k=1, then [tex]x=k^{-1/2}f(kt)[/tex] solves it for general k.
    Last edited: Sep 3, 2007
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