Boundary Value Problem + Green's Function

[benronan]

Boundary Value Problem + Green's Function
Consider the BVP

y''+4y=e^x
y(0)=0
y'(1)=0

Find the Green's function for this problem.



I am completely lost can someone help me out?

 

[yungman]

I am no expert, this looks like just an ODE problem where you only have one independent variable x:

1) Solve the associate homogeneous problem $y''+4y=0$ which is a second degree DE with constant coef.

2) Then use variation of parameter to get the particular solution.

3) $$y=y_c + y_p$$

4) Then use boundary condition to find the constant.

Just my 2 cents

 

[benronan]

yea but for a boundary condition should you have y(0) = c1 & y(t) = c2 , instead of y'(t) = c2

 

[yungman]

yea but for a boundary condition should you have y(0) = c1 & y(t) = c2 , instead of y'(t) = c2

I have not work through the problem, I am studying Green's function and that's the reason I look at this post.

I don't see from [itex]y''+4y=e^x[/tex] that it is even multi variables. Only independent variable is x. From my understanding, Green's function only deal with multi-variables. This is a simple 2nd degree non-homogeneous ODE with constant coef. with boundary condition.

 

[jackmell]

Hi. I'm not very familiar with Green's functions, but let me suggest a direction I think you need to go:

Consider the linear differential equation (of one variable):

$$Ly=f$$

where the L is the differential operator like in your case, it's $L=\frac{d^2}{dx^2}+4$. Then we can show the solution can be written in terms of a Green's function as:

$$y(x)=\int G(x,u)f(u)du$$

where:

$$G(x,u)=\sum_{n=1}^{\infty} \frac{\phi_n(x)\phi_n^{*}(u)}{\lambda_n}$$

where $\phi_n$ is a orthonormal set of eigenfunctions for the operator L, that is, normalized solutions to the equation:

$$y''+4y=\lambda_n y$$

subject to the given boundary conditions. So, just need to find those huh? Also, keep in mind the conjugate ($\phi^{*}$) of a real-valued function is just the function.

See: "Mathematics of Classical and Quantum Physics" by F. Byron and R. Fuller. Whole chapter on Green's functions.

 

[Coto]

I suggest taking a look at "Mathematical Physics" by Hassani or alternatively "Mathematical Methods for Physicists" by Arfken. Both have well developed sections on the use of Green's functions in solving ODEs.