Legendre differential equation and reduction of order

[CassieG]

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

I've included all the information from the question, any guidance in the right direction would be very welcome.

 

[jackmell]

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

$$y''+py'+qy=0$$

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

$$y_1 w'+(2y_1^'+p y_1)w=0$$

which you can solve via an integrating factor.

 

[CassieG]

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.