Scientific Computing with *Mathematica*®
pp 33-48 |
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# Linear ODEs with Constant Coefficients

## Abstract

**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*→ ∞.

## Keywords

Phase Portrait Constant Coefficient Nonlinear Differential Equation Stable Focus Unstable Node## Preview

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