Systems of First Order Equations

One of the fundamental concepts of analysis is that of a system of n simultaneous first order differential equations. If y 1(x), y 2(x),…, y n (x) are unknown functions of a single independent variable x, then the most general system of interest to us is one in which their derivatives y′ 1, y′2, …,y′ n are explicitly given as functions of x and y 1, y 2,…, yn : (1) y ′ 1 = f 1 ( x , y 1 , y 2 , … , y n ) y ′ 2 = f 2 ( x , y 1 , y 2 , … , y n ) ⋯ y ′ n = f n ( x , y 1 , y 2 , … , y n ) .