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Non-linear differential equations!

  1. Jul 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that w(t) = tanh(t) solves the nonlinear problem:

    w''(t)+2w(t)-2w3(t) = 0
    t ε ℝ

    2. Relevant equations
    [itex]\frac{d^2tanh(t)}{dt^2}[/itex] = -2tanh(t)sech2(t) = [itex]\frac{-8sinh(2t)cosh^2(t)}{(cosh(2t)+1)^3}[/itex]

    tanh(t) = [itex]\frac{sinh(2t)}{cosh(2t)+1}[/itex]


    tanh(t)3 = [itex]\frac{sinh^3(2t)}{(cosh(2t)+1)^3}[/itex]




    3. The attempt at a solution
    plug and chug? I'm not good at hyperbolic functions.

    Any ideas?
     
    Last edited: Jul 26, 2012
  2. jcsd
  3. Jul 26, 2012 #2
    yeah.. I keep getting stuck.

    currently at:

    [itex]\frac{[-6sinh(2t)cosh^2(t)+4cosh(2t)sinh(2t)+2sinh(2t)-2sinh^3(2t)]}{(cosh(2t)+1)^3}[/itex]
     
    Last edited: Jul 26, 2012
  4. Jul 26, 2012 #3

    HallsofIvy

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    It's very hard to decipher what you have written. Yes, if w= tanh(t) then [itex]w''= -2sech^2(t)tanh(t)[/itex] but it is not clear where you get the rest of that from. Just replacing w'' with [itex]-2sech^2(t)tanh(t)[/itex] and w with tanh(t), the left side becomes
    [itex]-2sech^2(t)tanh(t)+ 2tanh(t)- 2tanh^3(t)[/itex]
    Factor 2 tanh(t) out of that to get
    [itex]2tanh(t)(1- sech^2(t)- tanh^2(t))[/itex]

    Now, what is [itex]tanh^2(t)+ sech^2(t)[/itex]?
     
  5. Jul 26, 2012 #4
    Thank you! Also I was attempting to simplify in terms of sinh(t) and cosh(t).. I don't know why.
     
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