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A Second Order Differential Equation

  1. Oct 19, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi, this problem is from first chapter of Mathematical Methods of Physics by Mathews and Walker. (Problem 1-36, second edition)

    Consider the differential equation [itex]y'' - xy + y^3 = 0 [/itex] for large positive x.

    a-) Find an oscillating solution with two arbitrary constants.

    b-) Find a (non trivial) particular nonoscillating solution.


    2. Relevant equations

    [itex] x \rightarrow \infty [/itex] [itex] Ai(x) \rightarrow \frac{e^{-\frac{2}{3}x^{\frac{3}{2}}}}{2 \sqrt{\pi}x^{\frac{1}{4}}} [/itex] and [itex] Bi(x) \rightarrow \frac{e^{\frac{2}{3}x^{\frac{3}{2}}}}{\sqrt{\pi}x^{\frac{1}{4}}} [/itex]
    3. The attempt at a solution

    If we assume y is small then we get airy equation and [itex]Ai(x)[/itex] is a solution while [itex]Bi(x)[/itex] is not a solution since for [itex] x \rightarrow \infty { } Bi(x) \rightarrow \infty [/itex]. Also it should be noted that [itex]Ai(x)[/itex] does not have a root for large x therefore cannot be considered as an oscillated solution. Also WKB method didn't give me an oscillating solution but I can't say I know it well.

    Thanks
     
    Last edited: Oct 19, 2012
  2. jcsd
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