Question

Find the Green's function for the linear operator

Ly≡y′′+y relevant to the end conditions
y′(0)=y′(1)=0.
(b) Using Part (a), solve the boundary value problem
y′′+y+1=0,y′(0)=y′(1)=0.

Answer 1

To find the Green's function for the linear operator Ly≡y′′+y, solve the boundary value problem y′′+y+1=0 with the given end conditions y′(0)=y′(1)=0.

Step-by-step explanation:

In order to find the Green's function for the linear operator Ly≡y′′+y relevant to the end conditions y′(0)=y′(1)=0, we need to solve the boundary value problem y′′+y+1=0 with the same end conditions.

First, we need to find the general forms of the solutions in regions I and III, which can be verified to be of the form yI(x) = Aexp(-x) and yIII(x) = Aexp(x), respectively.

Then, we can find the solution in region II by solving the equation y′′+y+1=0 with the boundary conditions y'(0)=y'(1)=0. The solution in region II can be written as yII(x) = Bsinh(x) - Ccosh(x).