Green Function

[logistic_guy]

Determine Green’s Function and the solution for the following initial value problem:

\(\displaystyle u'' + \frac{1}{x}u' - \frac{1}{x^2}u = f(x), \ \ \ u(1) = 0, \ \ \ u'(1) = 1\)

 

[logistic_guy]

Let us first enjoy the taste of Green's Function structure!

\(\displaystyle g(x,s) = \frac{\Delta(x,s)}{a_0(s)W(s)}\)

 

[BeansNRice]

Congratulations, you have killed this board. I bet you feel pretty proud of yourself.

For your next feat you can go beat up a little girl. Go you.

 

[logistic_guy]

Congratulations, you have killed this board. I bet you feel pretty proud of yourself.

For your next feat you can go beat up a little girl. Go you.

What have you done so far?

Don't panic BeansNRice, we can solve this if you show us exactly where you're stuck at. I am not saying it's easy, but it's solvable!

Let us start with \(\displaystyle \Delta(x,s)\)

Since this Delta function depends on the homogeneous solution of the original differential equation,

we have to solve this first:

\(\displaystyle u'' + \frac{1}{x}u' - \frac{1}{x^2}u = 0\)

This is called Euler's differential equation. Have you solved this type of equations before?

The more information you tell us about what you know the better we can help you!