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Almost Kummer's Equation

  1. Jun 4, 2010 #1
    I met the following equation in my research, which is almost Kummer's equation (without the 2):

    x*y''+(b-2*x)*y'-a*y=0

    How can I represent this equation in terms of Kummer's? Or else, how solve it?
     
    Last edited: Jun 4, 2010
  2. jcsd
  3. Jun 7, 2010 #2
    Take your equation, and make the change of variable

    [tex]\tau = 2 x[/tex]

    This means that

    [tex]y^{\prime}_{x} = 2 y^{\prime}_{\tau} [/tex]

    and

    [tex]y^{\prime \prime}_{xx} = 4 y^{\prime \prime}_{\tau \tau} [/tex]

    Substitute these into your equation, and it becomes

    [tex]\tau y^{\prime \prime}_{\tau \tau} + (b - \tau) y^{\prime}_{\tau} - \frac{a}{2} y = 0[/tex]

    which is the hypergeomtric equation in the new variable.
     
  4. Jun 7, 2010 #3
    Perfect! Thank you.
     
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