Linear Algebra with Applications, 10th Edition

6.2 Systems of Linear Differential Equations

Eigenvalues play an important role in the solution of systems of linear differential equations. In this section, we see how they are used in the solution of systems of linear differential equations with constant coefficients. We begin by considering systems of first-order equations of the form

$$\begin{array}{cc}\hfill y_1^{\prime }& =a_{11}{y}_1+a_{12}{y}_2+\dots +a_{\text{1}n}{y}_n\hfill \\ \hfill y_2^{\prime }& =a_{21}{y}_1+a_{22}{y}_2+\dots +a_{2n}{y}_n\hfill \\ \hfill ⋮\\ \hfill y_n^{\prime }& =a_{n1}{y}_1+a_{n2}{y}_2+\dots +a_{nn}{y}_n\hfill \end{array}$$

where $y_i=f_i\left(t\right)$ is a function in $C^1[a,b]$ for each i. If we let

$$\begin{array}{ccc}\mathbf{Y}=\left[\begin{array}{c}{y}_1\\ y_2\\ ⋮\\ y_n\end{array}\right]& \text{and}& {\mathbf{Y}}^{\prime }=\left[\begin{array}{c}{y}_1^{\prime }\\ y_2^{\prime }\\ ⋮\\ y_n^{\prime }\end{array}\right]\end{array}$$

then the system can be written in the form

$${\mathbf{Y}}^{\prime }=A\mathbf{Y}$$

Y and ${\mathbf{Y}}^{\prime }$ are both vector functions of t. Let us consider the simplest case first. When $n=1$, the system is simply

$$y^{\prime }=ay$$ (1)

Clearly, any function of the form

$$\begin{array}{cc}y\left(t\right)=c{e}^{at}& (c\text{ an}\text{ arbitrary}\text{ constant ...}\end{array}$$