Another Green function question.

[MathematicalPhysicist]

find the eigenfunctions and eigen values of the next equation:
$$d^2y/dx^2+u_n^2y=0$$
where y(0)=0=y(pi).
Now find the green function of the above non-homogeneous equation, i.e:
$$d^2G_{\lambda}(x,a)/dx^2-\lambda G_{\lambda}(x,a)=\delta(x-a)$$
where a is in (0,pi) and lambda doesn't equal the -u_n^2.


Now here's what I did, the eigenfunction are y(x)=Asin(nx) and u_n=n.

Now for the green function for x different than a the above equation is 0, and the ( l is the same as lambda) solution is: G(x,a)=Ae^(sqrt(l)x)+Be^(-sqrt(l)x))
now in G(pi)=0=Ae^(2sqrt(l)pi)+B
B=-Ae^(2sqrt(l)pi) this is for x in (a,pi] for x in [0,a) we have B=-A, then we have yet without the a:
for x in [0,a) A(e^(sqrt(l)x)-e^(-sqrt(l)x))
for x in (a,pi] A(e^(sqrt(l)x)-e^(2sqrt(l)pi)e^(-sqrt(l)x))
now we need continuity at x=a, but from this we need to get that l=0, did I do something wrong in my reasoning here?

any advice on how to amend it?

 

[MathematicalPhysicist]

can anyone give some hint?