# First-order differential equations

• Martin Braun
Part of the Applied Mathematical Sciences book series (AMS, volume 15)

## 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
$$\frac{{dy}}{{dt}} = 3{y^{2}}\sin (t + y)$$
(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. For example, the function
$$y(t) = 2\sin t - \frac{1}{3}\cos 2t$$
is a solution of the second-order differential quation
$$\frac{{{d^{2}}y}}{{d{t^{2}}}} + y = \cos 2t$$
since
$$\frac{{{d^{2}}}}{{d{t^{2}}}}(2\sin t - \frac{1}{3}\cos 2t) + (2\sin t - \frac{1}{3}\cos 2t) = ( - 2\sin t + \frac{4}{3}\cos 2t) + 2\sin t - \frac{1}{3}\cos 2t = \cos 2t.$$
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## Keywords

Decimal Place White Lead Separable Equation Annual Interest Rate Orthogonal Trajectory
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