Legendre differential equation and reduction of order

  • Thread starter CassieG
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Homework Statement



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.

Homework Equations



See above.

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.
 
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Answers and Replies

  • #2
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You just need one solution to reduce the order so let y_1=x and y=y_1 v in your equation written as:

[tex]y''+py'+qy=0[/tex]

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.
 
  • #3
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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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