Green's Function on DE with variable coefficients

[jekyll]

Homework Statement

Hello. I'm taking a course on Mathematical Physics, based on Eugene Butkov's book. I'm having trouble solving a DE with variable coefficients to find Green's Function.

The problem asks to find Green's Function through direct construction.

Homework Equations

$$\frac{d}{dx} [ \frac{1}{x} \frac{dy}{dx} ] = f(x)$$
$$0 \leq x \leq 1$$
$$y(0) = y'(1) = 0$$

The Attempt at a Solution

I take it "through direct construction" means solving the system for G, replacing f(x) with the delta function. I've tried writing G(x,x') in series:

First, using a sine. The boundary conditions are satisfied, but when applying this G in the DE note that we get G'' and G', so there's a sine and a cosine and I can't use orthogonality to find the coefficients (that depend on x').

Then, I tried Frobenius method. And I get that all coefficients except for a0 are zero.

Any ideas on how to solve that equation? =(

 

[mighty2000]

Dear fellow student,

I understand your struggle with finding the Green's Function for a differential equation with variable coefficients. It can be a challenging task, but with the right approach, it is definitely achievable.

Firstly, I would suggest trying to solve the equation using the method of variation of parameters. This method involves finding a particular solution to the nonhomogeneous equation and then using it to construct the Green's Function. It may be helpful to review this method in your textbook or seek additional resources online.

Another approach you could try is using integral transforms, such as the Laplace transform or the Fourier transform. These transforms can help simplify the differential equation and make it easier to solve for the Green's Function.

Lastly, I would recommend seeking assistance from your instructor or a tutor if you are still having trouble. They can provide guidance and support to help you understand the concept better and find a solution to your problem.

I hope this helps and good luck with your studies!