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Periodic functions (or similar)

  1. Aug 10, 2011 #1
    are there non-connstant function that satisfy the following asumptions ??

    [tex] y(x)=y(kx) [/tex] they are 'periodic' but under DILATIONS

    and also satisfy the differential equation of the form (eigenvalue problem)

    [tex] axy'(x)+bx^{2}y''(x)=e_{n}y(x) [/tex]

    if the Lie Group is of translations [tex] y(x+1)=y(x) [/tex] we may have sine and cosine , however for the case of DILATIONS i do not know what functions can we take.
     
  2. jcsd
  3. Aug 10, 2011 #2

    hunt_mat

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    Homework Helper

    Your differential equation is of Euler type:
    [tex]
    bx^{2}y''+axy'-e_{n}y=0
    [/tex]
    Look for solutions of the for, [itex]y(x)=x^{n}[/itex], then:
    [tex]
    bn(n-1)x^{n}+anx^{n}-e_{n}x^{n}=0\Rightarrow (bn(n-1)+an-e_{n})x^{n}
    [/tex]
    So to obtain solutions we look for solutions of the quadratic:
    [tex]
    bn^{2}+(a-b)n-e_{n}=0
    [/tex]
    Which gives:
    [tex]
    n=\frac{b-a\pm\sqrt{(b-a)^{2}+4be_{n}}}{2b}
    [/tex]
     
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