Differential Equations and Their Applications pp 264-371 | Cite as

# Systems of differential equations

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## Abstract

In this chapter we will consider simultaneous first-order differential equations in several variables, that is, equations of the form A solution of (1) is since

$$\begin{gathered}
\frac{{d{x_1}}}{{dt}} = {f_1}\left( {t,{x_1},...,{x_n}} \right), \hfill \\
\frac{{d{x_2}}}{{dt}} = {f_2}\left( {t,{x_1},...,{x_n}} \right), \hfill \\
\vdots \hfill \\
\frac{{d{x_n}}}{{dt}} = {f_n}\left( {t,{x_1},...,{x_n}} \right). \hfill \\
\end{gathered} $$

(1)

*n*functions*x*_{1}(*t*),...,*x*_{n}(*t*) such that*dx*_{j}(*t*)/*dt*=*f*_{j}(*t, x*_{1}(*t*),...,*x*_{n}(*t*)),*j*= 1,2,...,*n*. For example,*x*_{1}(*t*) =*t*and*x*_{2}(*t*) =*t*^{2}is a solution of the simultaneous first-order differential equations$$\frac{{d{x_1}}}{{dt}} = 1{\kern 1pt} and{\kern 1pt} \frac{{d{x_2}}}{{dt}} = 2{x_1}$$

*dx*_{1}(*t*)/*dt*= 1 and*dx*_{2}(*t*)/*dt*= 2*t*= 2*x*_{1}(*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