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