ABSTRACT
“Linear” differential equations make up a rather large and important class of differential equations, both because they often arise in applications, and because there is an extensive theory and set of solution methods for these equations. There are two parts to this chapter. The first part begins by expanding the criteria for a first-order equation to be linear to the criteria for a second- and higher-order differential equation to be linear, and describing the difference between a “homogeneous” and “nonhomogeneous” equation — a difference that will play a major role later in the solving of linear differential equations. Existence and uniqueness results are also discussed here.
The second part of this chapter covers a method — reduction of order — for generating a lower-order linear differential equation from a linear equation, provided one solution to the original differential equation is already known. This is particularly useful when the original equation is of second order, since the resulting “reduced-order” equation can then be solved using methods for first-order equations, and the general solution to the original equation can then be easily generated.
