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# 10.2: Linear Systems of Differential Equations

• • William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathematics) at Trinity University
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$$\newcommand{\place}{\bigskip\hrule\bigskip\noindent} \newcommand{\threecol}{\left[\begin{array}{r}#1\\#2\\#3\end{array}\right]} \newcommand{\threecolj}{\left[\begin{array}{r}#1\$1\jot]#2\\[1\jot]#3\end{array}\right]} \newcommand{\lims}{\,\bigg|_{#1}^{#2}} \newcommand{\twocol}{\left[\begin{array}{l}#1\\#2\end{array}\right]} \newcommand{\ctwocol}{\left[\begin{array}{c}#1\\#2\end{array}\right]} \newcommand{\cthreecol}{\left[\begin{array}{c}#1\\#2\\#3\end{array}\right]} \newcommand{\eqline}{\centerline{\hfill\displaystyle#1\hfill}} \newcommand{\twochar}{\left|\begin{array}{cc} #1-\lambda\\#3-\lambda\end{array}\right|} \newcommand{\twobytwo}{\left[\begin{array}{rr} #1\\#3\end{array}\right]} \newcommand{\threechar}{\left[\begin{array}{ccc} #1-\lambda\\#4-\lambda\\#7 -\lambda\end{array}\right]} \newcommand{\threebythree}{\left[\begin{array}{rrr} #1\\#4\\#7 \end{array}\right]} \newcommand{\solutionpart}{\vskip10pt\noindent\underbar{\color{blue}\sc Solution({\bf #1})\ }} \newcommand{\Cex}{\fbox{\textcolor{red}{C}}\, } \newcommand{\CGex}{\fbox{\textcolor{red}{C/G}}\, } \newcommand{\Lex}{\fbox{\textcolor{red}{L}}\, } \newcommand{\matfunc}{\left[\begin{array}{cccc}#1_{11}(t)_{12}(t)&\cdots _{1#3}(t)\\#1_{21}(t)_{22}(t)&\cdots_{2#3}(t)\\\vdots& \vdots&\ddots&\vdots\\#1_{#21}(t)_{#22}(t)&\cdots_{#2#3}(t) \end{array}\right]} \newcommand{\col}{\left[\begin{array}{c}#1_1\\#1_2\\\vdots\\#1_#2\end{array}\right]} \newcommand{\colfunc}{\left[\begin{array}{c}#1_1(t)\\#1_2(t)\\\vdots\\#1_#2(t)\end{array}\right]} \newcommand{\cthreebythree}{\left[\begin{array}{ccc} #1\\#4\\#7 \end{array}\right]} 1 \ newcommand {\ dy} {\ ,\ mathrm {d}y} \ newcommand {\ dx} {\ ,\ mathrm {d}x} \ newcommand {\ dyx} {\ ,\ frac {\ mathrm {d}y}{\ mathrm {d}x}} \ newcommand {\ ds} {\ ,\ mathrm {d}s} \ newcommand {\ dt }{\ ,\ mathrm {d}t} \ newcommand {\dst} {\ ,\ frac {\ mathrm {d}s}{\ mathrm {d}t}}$$ A first order system of differential equations that can be written in the form \[\label{eq:10.2.1} \begin{array}{ccl} y'_1&=&a_{11}(t)y_1+a_{12}(t)y_2+\cdots+a_{1n}(t)y_n+f_1(t)\\ y'_2&=&a_{21}(t)y_1+a_{22}(t)y_2+\cdots+a_{2n}(t)y_n+f_2(t)\\ &\vdots\\ y'_n& =&a_{n1}(t)y_1+a_{n2}(t)y_2+\cdots+a_{nn}(t)y_n+f_n(t)\end{array}$

is called a linear system.

The linear system Equation \ref{eq:10.2.1} can be written in matrix form as

$\col{y'}n=\matfunc ann\col yn+\colfunc fn, \nonumber$

or more briefly as

$\label{eq:10.2.2} {\bf y}'=A(t){\bf y}+{\bf f}(t),$

where

$\bf y=\col yn,\quad A(t)=\matfunc ann,\quad \text{and} \quad{\bf f}(t)=\colfunc fn. \nonumber$

We call $$A$$ the coefficient matrix of Equation \ref{eq:10.2.2} and $${\bf f}$$ the forcing function. We’ll say that $$A$$ and $${\bf f}$$ are continuous if their entries are continuous. If $$\bf f={\bf 0}$$, then Equation \ref{eq:10.2.2} is homogeneous; otherwise, Equation \ref{eq:10.2.2} is nonhomogeneous.

An initial value problem for Equation \ref{eq:10.2.2} consists of finding a solution of Equation \ref{eq:10.2.2} that equals a given constant vector

$\bf k =\col kn. \nonumber$

at some initial point $$t_0$$. We write this initial value problem as

$\bf y'=A(t){\bf y}+{\bf f}(t), \quad {\bf y}(t_0)={\bf k}.\nonumber$

The next theorem gives sufficient conditions for the existence of solutions of initial value problems for Equation \ref{eq:10.2.2}. We omit the proof.

##### Theorem 10.2.1 : Existence

Suppose the coefficient matrix $$A$$ and the forcing function $${\bf f}$$ are continuous on $$(a,b)$$, let $$t_0$$ be in $$(a,b)$$, and let $${\bf k}$$ be an arbitrary constant $$n$$-vector. Then the initial value problem

$\bf y'=A(t){\bf y}+{\bf f}(t), \quad {\bf y}(t_0)= \bf k \nonumber$

has a unique solution on $$(a,b)$$.

##### Example 10.2.1
1. Write the system $\label{eq:10.2.3} \begin{array}{rcl} y_1'&=&\phantom{2}y_1+2y_2+2e^{4t} \\[4pt] y_2'&=&2y_1+\phantom{2}y_2+\phantom{2}e^{4t} \end{array}$ in matrix form and conclude from Theorem 10.2.1 that every initial value problem for Equation \ref{eq:10.2.3} has a unique solution on $$(-\infty,\infty)$$.
2. Verify that $\label{eq:10.2.4} {\bf y}= {1\over5}\twocol87e^{4t}+c_1\twocol11e^{3t}+c_2\twocol1{-1}e^{-t}$ is a solution of Equation \ref{eq:10.2.3} for all values of the constants $$c_1$$ and $$c_2$$.
3. Find the solution of the initial value problem $\label{eq:10.2.5} {\bf y}'=\left[\begin{array}{cc}{1}&{2}\\{2}&{1}\end{array} \right] {\bf y}+\twocol21e^{4t},\quad {\bf y}(0)={1\over5}\twocol3{22}.$

Solution a

The system Equation \ref{eq:10.2.3} can be written in matrix form as

${\bf y}'=\twobytwo1221{\bf y}+\twocol21e^{4t}.\nonumber$

An initial value problem for Equation \ref{eq:10.2.3} can be written as

${\bf y}'=\left[\begin{array}{cc}{1}&{2}\\{2}&{1}\end{array} \right] {\bf y}+\twocol21e^{4t}, \quad y(t_0)=\twocol{k_1}{k_2}. \nonumber$

Since the coefficient matrix and the forcing function are both continuous on $$(-\infty,\infty)$$, Theorem 10.2.1 implies that this problem has a unique solution on $$(-\infty,\infty)$$.

Solution b

If $${\bf y}$$ is given by Equation \ref{eq:10.2.4}, then

\begin{align*} A{\bf y}+{\bf f}&= {1\over5}\left[\begin{array}{cc}{1}&{2}\\{2}&{1}\end{array} \right]\twocol87e^{4t}+ c_1\left[\begin{array}{cc}{1}&{2}\\{2}&{1}\end{array} \right]\twocol11e^{3t} +c_2\left[\begin{array}{cc}{1}&{2}\\{2}&{1}\end{array} \right]\twocol1{-1}e^{-t} +\twocol21e^{4t}\\[4pt] &= {1\over5}\twocol{22}{23}e^{4t}+c_1\twocol33e^{3t}+c_2\twocol{-1}1e^{-t} +\twocol21e^{4t}\\[4pt] &= {1\over5}\twocol{32}{28}e^{4t}+3c_1\twocol11e^{3t}-c_2\twocol1{-1}e^{-t} \\[4pt] &={\bf y}'.\end{align*}

Solution c

We must choose $$c_1$$ and $$c_2$$ in Equation \ref{eq:10.2.4} so that

${1\over5}\twocol87+c_1\twocol11+c_2\twocol1{-1}={1\over5}\twocol3{22},\nonumber$

which is equivalent to

$\left[\begin{array}{cc}{1}&{1}\\{1}&{-1}\end{array} \right] \twocol{c_1}{c_2}=\twocol{-1}3.\nonumber$

Solving this system yields $$c_1=1$$, $$c_2=-2$$, so

${\bf y}={1\over5}\twocol87e^{4t}+\twocol11e^{3t}-2\twocol1{-1}e^{-t}\nonumber$

is the solution of Equation \ref{eq:10.2.5}.

##### Note

The theory of $$n \times n$$ linear systems of differential equations is analogous to the theory of the scalar n-th order equation $\label{eq:10.2.6} P_{0}(t)y^{(n)}+P_{1}(t)y^{(n-1)}+\cdots +P_{n}(t)y=F(t)$ as developed in Sections 9.1. For example by rewriting Equation \ref{eq:10.2.6} as an equivalent linear system it can be shown that Theorem 10.2.1 implies Theorem 9.1.1 (Exercise 10.2.12).