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Second Order differential equation involving chain rule

  1. Oct 25, 2013 #1
    1. The problem statement, all variables and given/known data
    Solve (d^2x)/(dt^2) = 2x(9 + x^2) given that dx/dt = 9 when x = 0 and x = 3 when t = 0


    2. Relevant equations



    3. The attempt at a solution

    v = dx/dt ....................... dv/dx = d^2x/dt^2

    dv/dx = v(dv/dx)

    v(dv/dx) = 18x +2x^3

    integrating and evaluating using limits and then you get

    (v^2/)2 - 81/2 = 9x^2 +.5x^4

    multiply by 2 and add 81 to both sides

    v^2 = 18x^2 + x^4 + 81


    dx/dt = v = + or - (18^2 + x^4 +81)^(1/2)

    this where I have a problem generally when we did these questions I would be able to just get the root without needing to use the square root symbol. I'm not sure how to integrate polynomials with a power.Does this require integration by substitution? Because I know its no longer apart of our course.

    Any help would be appreciated
     
  2. jcsd
  3. Oct 25, 2013 #2

    vela

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    ##x^4 + 18 x^2 + 81## is a perfect square.
     
  4. Oct 25, 2013 #3
    Oh right i didn't see that. So it would be ( x^2 + 9 )^2

    and then when you get the square root (x^2 + 9)
     
    Last edited: Oct 25, 2013
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