Consider the boundary value problem: \(y^{\prime \prime}-y=x, x \in(0,1)\), with boundary conditions \(y(0)=y(1)=0\).

a. Find a closed form solution without using Green's functions.

b. Determine the closed form Green's function using the properties of Green's functions. Use this Green's function to obtain a solution of the boundary value problem.

c. Determine a series representation of the Green's function. Use this Green's function to obtain a solution of the boundary value problem.

d. Confirm that all of the solutions obtained give the same results.