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B Which method to solve this 2nd order DE?

  1. Nov 5, 2016 #1
    I have ##\frac{d^2x(t)}{dt^2} + B(x(t))^3 = 0## for a system where I know the initial conditions and where B is a constant that's constructed from the properties of the system. I would like to find ##x(t)##.

    I've modelled the system in Python and produced some graphs. I know that ##x(t)## is some periodic function.

    Could somebody name the method that I should study so that I can get a solution?
     
  2. jcsd
  3. Nov 5, 2016 #2
    After some further reading I've got this...
    Let ##s = x'## so ##s' = x''##

    ##-Bx^3 = x'' = s' \frac{ds}{dt} = \frac{ds}{dx}\frac{dx}{dt} = \frac{ds}{dx} s##

    ##-\int{Bx^3 dx} = \int{s ds} ##

    ##\frac{-Bx^4}{4} + C = \frac{s^2}{2}##

    2C is just a constant that I'll rename C

    ##\sqrt{C-\frac{Bx^4}{2}} = s = \frac{dx}{dt}##

    Since ##x(0) = x_0## and ##x'(0) = 0## we have ##C-\frac{Bx_0^4}{2} = 0## so ##C = \frac{Bx_0^4}{2}##

    ##\frac{dx}{dt} = \sqrt{\frac{B}{2}}\sqrt{x_0^4-x^4}##

    ##\sqrt{\frac{B}{2}}\int{\frac{1}{\sqrt{x_0^4-x^4}}dx} = \int{dt}##

    I don't know if this is correct, but it seems plausible. Now to evaluate the integral.
     
  4. Dec 8, 2016 #3

    joshmccraney

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    Gold Member

    Could you state the initial conditions? I ask because you may be able to perturb the equation, take a naive expansion, and approximating your solution analytically. It would not be exact but the method can be incredibly close (like a truncated Taylor series).
     
  5. Dec 8, 2016 #4

    lurflurf

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    Homework Helper

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