General Properties of the Differential Equations

Abstract

Consider a system of differential equations:

$$ \frac{{{\text{d}}x_{1} }}{{{\text{d}}t}} = X_{1} ,\quad \frac{{{\text{d}}x_{2} }}{{{\text{d}}t}} = X_{2} ,\ldots \frac{{{\text{d}}x_{n} }}{{{\text{d}}t}} = X_{n} , $$

(1)

where $t$ represents the independent variable that we will call time and $x_{1} ,x_{2} ,\; \ldots x_{n}$ the unknown functions, where $X_{1} ,X_{2} , \ldots \,X_{n}$ are given functions of $x_{1} ,x_{2} , \ldots \,x_{n}$. We suppose in general that the functions $X_{1} ,X_{2} , \ldots \,X_{n}$ are analytic and one-to-one for all real values of $x_{1} ,x_{2} , \ldots \,x_{n}$.