Systems of Differential Equations

Abstract

In many (in fact, most) cases, differential equations of the form $x' = f\left( {t,\left. x \right)} \right.$ are inadequate for the description of a physical system; more variables are needed to specify its state at any time t. Usually a state of the physical system will be specified by the values of several functions, ${x_i}\left( t \right)$, for i 1,2…,n. If we know “forces” giving for each of them the derivative ${{x'}_i}\left( t \right)$ with respect to time, in terms of the values of all the variables (at that particular time) and perhaps also of time, then the evolution of the physical system can be described by a system of n first order differential equations,

$${{x'}_i} = f\left( {t,{x_1}} \right.,{x_2},...,\left. {{x_n}} \right)$$

((1))

for i= 1, 2,…, n, or as a single first order differential equation in ℝⁿ, where the vectors arc- usually written in print as boldface, or by hand with arrows; i.e.,

$$x'\left( t \right) = f\left( {t,\left. x \right)} \right. or \vec x' = \vec f\left( {t,} \right.\left. {\vec x} \right)$$

((2))