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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: Wiley-Interscience, 1979. ISBN: 0471819670.
Gustafson, K. "Fourier Series and Hilbert Space." Chapter 2 in Introduction to Partial Differential Equations and Hilbert Space Methods. Mineola, New York: Dover Publications, 1999. ISBN: 0486612716.
Solving Linear Ordinary Differential Equations (PDF)