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10: Systems of Linear Differential Equations

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Reference: Boyce and DiPrima, Chapter 7

Here, we consider the simplest case of a system of two coupled homogeneous linear first-order equations with constant coefficients. The general system is given by

˙x1=ax1+bx2,˙x2=cx1+dx2,

or in matrix form as

ddt(x1x2)=(abcd)(x1x2).

The short-hand notation will be

˙x=Ax.

Although we can write these two first-order equations as a single second-order equation, we will instead make use of our newly learned techniques in matrix algebra. We will also introduce the important concept of the phase space, and the physical problem of coupled oscillators.


This page titled 10: Systems of Linear Differential Equations is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Jeffrey R. Chasnov via source content that was edited to the style and standards of the LibreTexts platform.

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