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Legendre differential equation and reduction of order

  1. Dec 7, 2011 #1
    1. The problem statement, all variables and given/known data

    Question is to find a general solution, using reduction of order to:

    (1-x^2)y" - 2xy' +2y = 0

    (Legendre's differential equation for n=1)

    Information is given that the Legendre polynomials for the relevant n are solutions, and for n=1 this means 'x' is a solution.

    2. Relevant equations

    See above.

    3. The attempt at a solution

    'x' is one solution, I need another to form the general solution. I tried solving the characteristic equation in terms of x, as shown at this link. http://www.bravus.com/Legendre.jpg [Broken]

    I've included all the information from the question, any guidance in the right direction would be very welcome.
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Dec 7, 2011 #2
    You just need one solution to reduce the order so let y_1=x and y=y_1 v in your equation written as:


    do all that substituting and letting v'=w you should get:

    [tex]y_1 w'+(2y_1^'+p y_1)w=0[/tex]

    which you can solve via an integrating factor.
  4. Dec 7, 2011 #3
    That makes a lot of sense. Just working through it now, but I think it was realising I *had* reduced the order and had a first order ODE and then solving with the integrating factor was the point I was missing.

    Thanks very much.
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