With no disrespect to my professors in undergraduate and graduate school, I do not seem to recall spending much time on the topic of solving non-homogeneous, linear boundary value problems. Even as a professor of mathematics myself, I do not usually have the time during the semester to discuss boundary value problems in our ODE course. I feel plenty of time is spent on the method of undetermined coefficients and variation of parameters for solving second order non-homogeneous differential equations. I recently had the need to organize some notes on using Green’s functions to solve non-homogeneous initial and boundary value ordinary differential equations and initial, boundary value partial differential equations. If you are interested in seeing the method developed and applied to nth order, linear, non-homogeneous boundary value problems on intervals of the form [a, b], check out these notes. I have attempted to concisely motivate the method, so that it can be immediately employed in examples. Two examples are included.
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