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I Determining the coefficient of the legendre polynomial

  1. May 1, 2017 #1
    We know that the solution to the Legendre equation:
    $$ (1-x^2)\frac{d^2 y}{dx^2} - 2x \frac{dy}{dx} + n(n+1) = 0 $$
    is the Legendre polynomial $$ y(x) = a_n P_n (x)$$

    However, this is a second order differential equation. I am wondering why there is only one leading coefficient. We need two conditions to determine the unique solution to a 2nd order ODE, don't we?
     
  2. jcsd
  3. May 1, 2017 #2

    Orodruin

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    In addition to the Legendre polynomials, there is a second linearly independent solution to the differential equation, usually denoted ##Q_n(x)##. These are usually thrown away as they are singular as ##x\to\pm 1## and a typical requirement is that the functions are regular in these limits.
     
  4. May 1, 2017 #3
    That solved my puzzle! Thank you!
     
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