# Green function

1. Apr 28, 2007

### sensitive

1. The problem statement, all variables and given/known data

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?

2. Apr 28, 2007

### HallsofIvy

Staff Emeritus
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= xm, then y'= mxm-1 and y"= m(m-1)xm-2. Putting that into the equation, Lx= 0, you get
x2(m(m-1)xm-2- 2x(mxm-1)+ 2xm=- m(m-1)xm- 2m xm+ 2xm= 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.

3. Apr 28, 2007

### 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?