First-Order Differential Equations

Introduction

In this chapter, we study first-order differential equations. The standard form of a first-order differential equation is ¹

$$\begin{array}{}\text{(A)}& \frac{dy}{dx}=f(x,y)\end{array}$$

where $y$ is the unknown function of $x$ and $f$ is a given function defined on a region $D$ in the $xy$-plane.

If $f(x,y)$ is written as a quotient of two functions $M(x,y)$ and $N(x,y)$, namely

$$f(x,y)=-\frac{M(x,y)}{N(x,y)}$$

then (A) becomes ${\displaystyle \frac{dy}{dx}=-\frac{M(x,y)}{N(x,y)},}$ which is equivalent to the differential form

$$\begin{array}{}\text{(B)}& M(x,y)dx+N(x,y)dy=0.\end{array}$$

If we write a differential equation in the form (B) either $x$ or $y$ can be regarded as the independent variable.

Footnotes

1. Although (A) does not cover all first-order differential equations, it is inclusive enough for almost all applications. The most general form for a first-order differential equation is $F(x,y,y^{\prime })=0$. If we assume we can solve for $y^{\prime }$ (which is not always true), we will obtain $y^{\prime }=f(x,y)$. ↩