First Order Differential Equations Systems

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, ẋ = 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))