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2nd order non-linear homogeneous differential equation

  1. Jul 26, 2012 #1
    1. The problem statement, all variables and given/known data
    Find a solution (Z2) of:
    z'' + 2z - 6(tanh(t))2z = 0

    that is linearly independent of Z1 = sech2


    2. Relevant equations



    3. The attempt at a solution
    reduction of order gives you

    v''(t)(Z1(t))+v'(t)(2 * Z1'(t)) + v(t)(Z1''(t)+p(t)Z1'(t)) = 0
    however the third term on the LHS can be dropped since we know that Z1 is a solution to the original problem.

    v''(t)(Z1(t))+v'(t)(2 * Z1'(t)) = 0 = sech2(t)v''(t) + 2(-2tanh2(t)sech2(t))v'(t)

    let y = v'

    sech2(t)y'(t) + 2(-2tanh2(t)sech2(t))y(t) = 0

    divide both sides by sech2(t)

    y'(t) - 4tanh2(t)y(t) = 0

    from here would I use integrating factor, or should I have done exact equations for the step before this?

    using integrating factor
    μ(t) = e(4tanh(t)-4t)
    y = e-(4tanh(t)-4t)
    v = ∫e-(4tanh(t)-4t)dt

    can any1 point me in the correct direction? I also don't know how to integrate the last part..
     
    Last edited: Jul 26, 2012
  2. jcsd
  3. Jul 26, 2012 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member


    Check the derivative of sech2(t).

    ehild
     
  4. Jul 27, 2012 #3
    ahh that works beautifully. thanks ehild.
     
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