6.1. Introduction

A differential equation is a mathematical expression including one or more derivatives of an unknown function. A differential equation’s order is determined by the highest derivative it includes. If a differential equation involves an unknown function of just one variable, it is called an ordinary differential equation; if it comprises partial derivatives of more than one variable, it is called a partial differential equation. For the time being, we will just discuss ordinary differential equations, which we will simply refer to as differential equations [Barbu, 2016, Trench, 2001].

First-order differential equations of the type are the simplest differential equations, and they can be expressed in the following form [Barbu, 2016, Trench, 2001]

(6.1)\[\frac{dy}{dt}=f(t) \mbox{\quad or, equivalently, \quad}y'=f(t),\]

\(f\) is a known function of \(t\). We learned how to identify functions that satisfy this sort of problem in Calculus and Differential Equation classes.

Example: For example, if

\[\begin{equation*} y'=t^3, \end{equation*}\]

then

\[\begin{equation*} y=\int t^3\, dt=\frac{t^4}{4}+c, \end{equation*}\]

where \(c\) is an arbitrary constant.

It is not necessary to study differential equations like (6.1) except for illustration reasons in this section. We’ll generally look at differential equations that look like this:

(6.2)\[\begin{equation} \label{eq:1.2.2} y^{(n)}=f(t,y,y', \dots,y^{(n-1)}), \end{equation}\]

where at least one of the functions \(y\), \(y'\), \dots, \(y^{(n-1)}\) occurs on the right [Trench, 2001].

Examples:

\[\begin{align*} \begin{array}{rcll} \displaystyle{dy\over dt} - t^2&=&0&\mbox{(first order)},\\ \displaystyle{dy\over dt}+3ty^3&=&-2&\mbox{(first order)},\\ \displaystyle{d^2y\over dt^2}+2\displaystyle{dy\over dt}+y&=&4t&\mbox{(second order)},\\ ty'''+y^2&=&\sin(t) &\mbox{(third order)},\\ y^{(n)}+ty'+3y&=&t&\mbox{($n$-th order)}. \end{array} \end{align*}\]