First-order differential equations

Abstract

This book is a study of differential equations and their applications. A differential equation is a relationship between a function of time and its derivatives. The equations

$$\frac{{dy}}{{dt}}\, = \,3{y^2}\,\sin \left( {t\, + \,y} \right)$$

(i)

and

$$\frac{{{d^3}y}}{{d{t^3}}}\, = \,e{\,^{ - y}}\, + \,t\, + \,\frac{{{d^2}y}}{{d{t^2}}}$$

(ii)

are both examples of differential equations. The order of a differential equation is the order of the highest derivative of the function y that appears in the equation. Thus (i) is a first-order differential equation and (ii) is a third-order differential equation. By a solution of a differential equation we will mean a continuous function y(t) which together with its derivatives satisfies the relationship.