The following materials are recommended for background reading and study.
Stakgold, I. "Green's Functions: Intuitive Ideas." Chapter 1 in Green's Functions and Boundary Value Problems. New York, NY: Wiley-Interscience, 1979. ISBN: 9780471819677.
Gustafson, K. "Fourier Series and Hilbert Space." Chapter 2 in Introduction to Partial Differential Equations and Hilbert Space Methods. New York, NY: Dover Publications, 1997. ISBN: 9780486612713.
Solving Linear Ordinary Differential Equations (PDF)