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Help in solving a second-order linear differential equation

  1. Dec 6, 2006 #1
    [tex]
    \frac{{d^2 y}}{{dx^2 }} + \left( {Ax + B} \right)y = 0
    [/tex]

    I have tried lots of substitions, but a solution won't pop out. Can anyone help solve this?

    Thanks.
     
  2. jcsd
  3. Dec 7, 2006 #2
    Make the change of variable

    [tex]Ax + B = \lambda u [/tex]

    (lambda is a constant) this will give you

    [tex]\frac{A^2}{\lambda^2} \frac{d^2 y}{d u^2} + \lambda u y = 0 [/tex]

    so if you then set

    [tex] \lambda = -A^{\frac{2}{3}} [/tex]

    you then have

    [tex] \frac{d^2 y}{d u^2} - u y = 0 [/tex]

    which is the Airy equation (in u). Have a look on Wikipedia or elsewhere on Airy functions and such - or just type in "Airy Equation".

    edit: note you will actually get three different solutions as the condition for lambda is

    [tex] \frac{\lambda^3}{A^2} = -1 [/tex]

    which means that there are three values of lambda that satisfy this (i.e. three distinct cube-roots) - one will be real-valued (already given) plus two complex ones.

    Here is a link for the Airy function

    http://mathworld.wolfram.com/AiryFunctions.html
     
    Last edited: Dec 8, 2006
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