(Q4) Derive the Green's function for
with boundary conditions as
First note that the solution is almost already is its self adjoint form.
So we need to find two different solutions such that each one satisfies one of the Boundary Conditions.
As this gives and
As this gives
So the Green's function has the form
Now we use the Wronskian and the and the condiditon that
so from above we have that and
Now using this we get
Now we can pick and to be any real numbers that satisfy the above equation for example
This gives