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Systems of Differential Equations

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

Abstract

In this chapter we will consider simultaneous first order differential equations in several variables, that is, equations of the form
$$ \begin{gathered} \frac{{d{x_1}}}{{dt}} = {f_1}\left( {t,{x_1},...,{x_n}} \right), \hfill \\ \frac{{d{x_2}}}{{dt}} = {f_2}\left( {t,{x_1},...,xn} \right),...,\frac{{d{x_n}}}{{st}} = {f_n}\left( {t,{x_1},...,{x_n}} \right) \hfill \\ \end{gathered} $$
(1)
A solution of Equation (1) is n functions xl(t),…,xn(t) such that \( \frac{{d{x_j}\left( t \right)}}{{dx}} = {f_j}\left( {t,{x_1}\left( t \right),...,{x_n}\left( t \right)} \right),j = 1,2,...,n \) For example, xl(t) = t and x2(t) = t2 is a solution of the simultaneous first order differential equations \( \frac{{d{x_1}}}{{dt}} = 1 \) and \( \frac{{d{x_2}}}{{dt}} = 2{x_1} \) Since \( \frac{{d{x_1}^{\left( t \right)}}}{{dt}} = 1 \) and \( \frac{{d{x_2}^{\left( t \right)}}}{{dt}} = 2t = 2{x_1}\left( t \right) \) dt

Keywords

Vector Space Linear Transformation Characteristic Polynomial Scalar Multiplication Independent Solution 
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 1975

Authors and Affiliations

  • Martin Braun
    • 1
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA

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