The reduction of order method of the previous chapter is limited in value, and other more useful methods for solving homogeneous linear differential equations are developed in future chapters. However, the use of these methods requires an understanding of the nature of general solutions to these equations. Developing that understanding is the goal of this chapter. Here, the basic “theory” of general solutions to homogeneous linear differential equations of any order is developed and illustrated (with particular attention given to second-order equations). The concepts of “linear combinations”, “superposition”, “linear (in)dependence” and “fundamental sets” of solutions are developed and their relations with general solutions is demonstrated. The discussion culminates with theorems on superposition and on constructing general solutions to homogeneous linear differential equations and constructing corresponding solutions to related initial-value problems. These theorems will play a fundamental role throughout most of this text.

Finally, one major difficulty in determining whether a given set of solutions is a fundamental set is addressed by the development of the “Wronskians”.