# First-order differential equations

Chapter

## 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 and are both examples of differential equations. The order of a differential equation is the order of the highest derivative of the function is a solution of the second-order differential quation since .

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

(i)

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

(ii)

*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$$ y(t) = 2\sin t - \frac{1}{3}\cos 2t $$

$$ \frac{{{d^{2}}y}}{{d{t^{2}}}} + y = \cos 2t $$

$$ \frac{{{d^{2}}}}{{d{t^{2}}}}(2\sin t - \frac{1}{3}\cos 2t) + (2\sin t - \frac{1}{3}\cos 2t) = ( - 2\sin t + \frac{4}{3}\cos 2t) + 2\sin t - \frac{1}{3}\cos 2t = \cos 2t. $$

## Keywords

Decimal Place White Lead Separable Equation Annual Interest Rate Orthogonal Trajectory
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## Copyright information

© Springer-Verlag New York, Inc. 1983