Systems of Ordinary Differential Equations

A system of ordinarydifferential equations (briefly, ODE’s) is a problem of finding functions $$y_{1}(x),\ldots,y_{n}(x)$$ on some open interval in $$\mathbb{R}$$ such that 1.1.1 $$\displaystyle{ y_{k}\prime(x) = f_{k}(x,y_{1}(x),\ldots,y_{n}(x))\quad \text{for}\quad k = 1,\ldots,n }$$ where f k are continuous functions of n + 1 real variables. Note that then y i , since they are required to have a derivative, must in particular be continuous, and the derivative is then also continuous by (1.1.1). The expression “ordinary” indicates that there appear only derivatives of functions of one variable, not partial derivatives of functions of several variables.