Where Do Rayleigh-Ritz Trial Functions for Laguerre Equation Come From?

[Lengalicious]

Homework Statement

I need to figure out the trial function for the Laguerre equation,

$$xy''+(1-x)y'+ny=0$$

Homework Equations

The Attempt at a Solution

I have found multiple examples of the Rayleigh Ritz method on google, however, I am completely dumbfounded as to where the trial functions are coming from, my professor is making an absolute meal out of variational methods in calculus and not managing to explain anything well at the same time.

Any hints or references to somewhere which might be able to explain better would be great.

I have gotten Laguerres equation into sturm-liouville form and I know that generally the trial function is $$y(x)=\sum_{i=0}^{\infty}c_iu_i$$ but that really isn't helping me figure out where example specific trial functions are coming from.

 

[Chestermiller]

If I remember correctly, when you are using the Rayleigh Ritz method, don't you choose the trial functions so that they satisfy the boundary conditions exactly?

Chet