1. Solve homogeneous equation
We note to ensure linear independence of solutions as the Wronskian is .
3. continuous at
Therefore Eq. 1,
4. Jump Condition
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!