Systems of Differential Equations

Abstract

Let $y_{1}(t),\,y_{2}(t),\ldots,y_{n}(t)$ are unknown functions of a single independent variable t, the most interesting systems in applications are systems of the form

$$\displaystyle\begin{cases}y_{1}^{\prime}(t)=f_{1}(t,y_{1}(t),y_{2}(t),\ldots,y_{n}(t)),\\ y_{2}^{\prime}(t)=f_{2}(t,y_{1}(t),y_{2}(t),\ldots,y_{n}(t)),\\ \;\vdots\\ y_{n}^{\prime}(t)=f_{n}(t,y_{1}(t),y_{2}(t),\ldots,y_{n}(t)),\end{cases}$$

(6.1)

where we have n dependent variables $y_{1},y_{2},\ldots,y_{n}$ and one independent variable t. Later on, we may drop the t in order to shorten the notation.

It is not hard to see that any single differential equation of nth order of the form

$$\displaystyle y^{(n)}=f\Big(t,y,y^{\prime},y^{\prime\prime},\ldots,y^{(n-1)}\Big)$$

(6.2)

can be written as system of first order differential equations of the form (6.1). Indeed, we introduce the new variables

$$\displaystyle y_{1}=y,\quad y_{2}=y^{\prime},\quad\ldots,\quad y_{n}=y^{(n-1)},$$

(6.3)

then we obtain

$$\displaystyle\begin{cases}y_{1}^{\prime}=y_{2},\\ y_{2}^{\prime}=y_{3},\\ \;\vdots\\ y_{n}^{\prime}=f(t,y_{1},y_{2},\ldots,y_{n}).\end{cases}$$

(6.4)

More detailed discussion on the linear version of (6.2) is given in 6.2.7.