Construct Green’s function for the following boundary value problem:

with y(0) = y(1) = 0.

To construct the Green's function for the given boundary value problem, we first need to solve the homogeneous differential equation:

The characteristic equation of the above differential equation is:

The roots of this equation are:

r = ±3i

Therefore, the general solution of the homogeneous differential equation is:

y_h(x) = c_1 cos(3x) + c_2 sin(3x)

To satisfy the boundary conditions, we need to have y_h(0) = y_h(1) = 0. From the first boundary condition, we get:

c_1 = 0

From the second boundary condition, we get:

c_2 sin(3) = 0

Since sin(3) is not equal to 0, we must have:

c_2 = 0

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