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Systems of differential equations

  • Martin Braun
Chapter
Part of the Texts in Applied Mathematics book series (TAM, volume 11)

Abstract

In this chapter we will consider simultaneous first-order differential equations in several variables, that is, equations of the form
$$\begin{gathered} \frac{{d{x_1}}}{{dt}} = {f_1}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \frac{{d{x_2}}}{{dt}} = {f_2}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \vdots \hfill \\ \frac{{d{x_n}}}{{dt}} = {f_n}\left( {t,{x_1},...,{x_n}} \right). \hfill \\ \end{gathered} $$
(1)
A solution of (1) is n functions x1(t),..., xn(t) such that dxj(t)/dt = fj(t, x1(t),..., xn(t)), j = 1,2,..., n. For example, x1(t) = t and x2(t) = t2 is a solution of the simultaneous first-order differential equations
$$\frac{{d{x_1}}}{{dt}} = 1{\kern 1pt} and{\kern 1pt} \frac{{d{x_2}}}{{dt}} = 2{x_1}$$
since dx1(t)/dt = 1 and dx2(t)/dt = 2t = 2x1(t).

Keywords

Vector Space Linear Transformation Scalar Multiplication Independent Solution Nonzero Vector 
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 Science+Business Media New York 1993

Authors and Affiliations

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

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