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

Keywords

Decimal Place White Lead Separable Equation Annual Interest Rate Orthogonal Trajectory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 1983

Authors and Affiliations

  • Martin Braun
    • 1
  1. 1.Department of Mathematics, Queens CollegeCity University of New YorkFlushingUSA

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