# 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
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