Discontinuous Functions and the Laplace Transform

Abstract

Our focus in this chapter is a study of first and second order linear constant coefficient differential equations

$$ \begin{array}{rlrlrl} y\prime + ay & = f(t), & & \\ y\prime\prime + ay\prime + by & = f(t), & & \end{array} $$

where the input or forcing function f(t) is more general than we have studied so far. These types of forcing functions arise in applications only slightly more complicated than those we have already considered. For example, imagine a mixing problem (see Example 11 of Sect. 1.4 and the discussion that followed it for a review of mixing problems) where there are two sources of incoming salt solutions with different concentrations as illustrated in the following diagram.