Finding the Form of L: An Example

[sensitive]

Homework Statement

An example in my notes.

To solve for green function of a differential equation.
eg x^2(y'') - 2x(y') + 2y = x^3ln(x)

The first thing is to solve the kernel by solving Ly = 0
and it says the form of L suggests trying y = x^m

How to I obtain the suggested form of L?

 

[HallsofIvy]

By taking the suggestion?
You are told to consider the equation Ly= 0, the "associated homogeneous" equation to your non-homgeneous equation. Since you are given the equation x²y"- 2xy'+ 2y= x³ln x, it follows that Lx= x²y"- 2xy'+ 2y= 0.


If y= x^m, then y'= mx^m-1 and y"= m(m-1)x^m-2. Putting that into the equation, Lx= 0, you get
x²(m(m-1)x^m-2- 2x(mx^m-1)+ 2x^m=- m(m-1)x^m- 2m x^m+ 2x^m= 0 so either x=0 or m(m-1)- 2m+ 2= m²- 3m+ 2= (m-2)(m-1)= 0. m= 1 or m= 2 which gives you the general solution y= Cx+ Dx².

 

[sensitive]

Thx... I understand how to get the general solution but how to I come up with a suggested form?

Say i was given a differential equation. For example Ly = y'' - alpha^2*y = h(x). So first i need to make the equation to Ly = 0. But what is the suggested form?