Linear ODEs with Constant Coefficients

Abstract

In the previous chapter a fundamental theorem was introduced assuring existence, uniqueness, and continuity with respect to the initial conditions of the solution of an initial-value problem relative to Equation (1.3). Another fundamental problem is to find the explicit form of the solution. However, this is possible in only a few cases; in the other cases, many other strategies can be pursued to collect information on the solutions. In this chapter the linear differential systems with constant coefficients are analyzed, that is, systems with the form

$$ \dot x = Ax + b(t), $$

where A is an n × n constant matrix, and b(t) is a known column vector. This class is important for the following reasons:

• 1. It is possible to exhibit the general integral in a closed form.

• 2. In Chapter 1 a few simple examples showed that the mathematical modeling leads to a more or less difficult differential equation. A (scalar or vector) first-order differential equation is nothing but a relation between the unknown and its derivative so that the simplest models follow from the assumption that this relation is linear. Often, this equation represents a first approximation of more accurate descriptions, which usually lead to nonlinear differential equations. For example, in Chapter 1, a linear equation was obtained by attempting to describe the population growth in the absence of any constraint. When the constraint deriving from the existence of an upper bound M for the number of individuals constituting the population was taken into account, the nonlinear logistic equation was derived. It is plain to verify that this equation reduces to (1.1) when M → ∞.