1. Solve homogeneous equation

2. for

We note to ensure linear independence of solutions as the Wronskian is .

Hence

3. continuous at

ThereforeEq. 1,

4. Jump Condition

Which gives:

Eq. 2

Now Eq. 1 and Eq. 2 together give:

Now using Eq. 1 again, we know

Also we can simplify:

So putting this all together, we get our Green's function:

From here you should be able to represent the solution as a sum of two integrals that utilize the Green's function. Good luck

P.S. Let me know if I made any mistakes!