# First Order Differential Equations Systems

Chapter

## Abstract

A typical system of n first order differential equations is of the form
$$\dot{x}(t) = A(t)x(t) + b(t); x(0) = {{x}_{0}}$$
(5.1)
when A(t) is in general an n × n time variant coefficient matrix and b(t) a time variant n-vector. The constant coefficient case emerges as a particular one in which A and b are constant. The system (5.1) is homogeneous if b = 0 and non-homogeneous if b ≠ 0. The solution of the homogeneous part, = Ax is called the general solution of the complementary function, x c (t) and the solution that fits (5.1) is called the particular integral (x p ) or equilibrium solution (x e ). The combination of the two, x(t) = x c (t) + x e , gives the complete solution of (5.1). In general, if vectors x1, x2,…, xn are each a solution of (5.1), so is their linear combination
$$x\left( t \right) = {c_1}{x^1}\left( t \right) + {c_2}{x^2}\left( t \right) + \cdots + {c_n}{x^n}\left( t \right).$$
(5.2)

## Keywords

Fundamental Matrix Complex Root Complex Eigenvalue Price Vector Jordan Canonical Form
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.