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## Introduction to Systems of Linear ODEs

Suppose $$x_1,x_2,\ldots,x_n$$ are $$n$$ functions of $$t$$. Consider the following system of $$n$$ linear ODEs: $\begin{eqnarray*} \begin{array}{rccccccccc} x_1'=& a_{11}(t)x_1&+&a_{12}(t)x_2&+&\cdots &+&a_{1n}(t)x_n&+&g_1(t)\\ x_2'=& a_{21}(t)x_1&+&a_{22}(t)x_2&+&\cdots &+&a_{2n}(t)x_n&+&g_2(t)\\ \vdots&\vdots&&\vdots&& &&\vdots&&\vdots\\ x_n'=& a_{m1}(t)x_1&+&a_{m2}(t)x_2&+&\cdots &+&a_{mn}(t)x_n&+&g_n(t).\\ \end{array} \end{eqnarray*}$ It can be simply written in the following matrix form: $$$\overrightarrow{x}'=A\overrightarrow{x}+\overrightarrow{g}, \;\;\;\;(50)$$$ where $$A=\left[\begin{array}{cccc} a_{11}&a_{12}&\cdots &a_{1n}\\ a_{21}&a_{22}&\cdots &a_{2n}\\ \vdots&\vdots& &\vdots\\ a_{n1}&a_{n2}&\cdots &a_{nn} \end{array}\right],\; \overrightarrow{x}=\left[\begin{array}{c} x_1\\x_2\\ \vdots\\x_n\end{array} \right], \mbox{ and } \overrightarrow{g}=\left[\begin{array}{c} g_1\\g_2\\ \vdots\\g_n \end{array} \right].$$ $$A$$ is called the coefficient matrix of (50). When $$\overrightarrow{g}=\overrightarrow{0}$$, (50) is a homogeneous system. Similarly when $$\overrightarrow{g}\neq\overrightarrow{0}$$, (50) is a nonhomogeneous system

Example. A multiple mass-spring system can have the following equations of motion: $\begin{eqnarray*} m_1x_1''&= &-k_1x_1+k_2(x_2-x_1)+F_1(t)\\ m_2x_2''&= &-k_2(x_2-x_1)+F_2(t). \end{eqnarray*}$
Example. Write the following system in matrix form. $\begin{array}{ccrcrcr} x_1' &=& x_1 &+& x_2 &+& e^t\\ x_2' &=& 4x_1 &-& 2x_2 &+& t^2 \end{array}$ Solution. $\left[\begin{array}{c} x_1'\\x_2'\end{array} \right] =\left[\begin{array}{c}x_1+x_2+e^t\\4x_1-2x_2+t^2\end{array} \right] =\left[\begin{array}{rr} 1&1\\4&-2\end{array} \right]\left[\begin{array}{c} x_1\\x_2\end{array} \right] +\left[\begin{array}{c} e^t\\t^2\end{array} \right]$ $\text{i.e., } \overrightarrow{x}'=A\overrightarrow{x}+\overrightarrow{g},$ where $$A=\left[\begin{array}{rr} 1&1\\4&-2\end{array} \right],\; \overrightarrow{x}=\left[\begin{array}{c} x_1\\x_2\end{array} \right], \mbox{ and } \overrightarrow{g}=\left[\begin{array}{c} e^t\\t^2\end{array} \right].$$

Example. Write the following IVP in matrix form. $\begin{eqnarray*} &\begin{array}{ccrcr} x_1' &=& 4x_1 &-& 3x_2\\ x_2' &=& 8x_1 &-& 6x_2 \end{array}\\ &x_1(0)=5,\; x_2(0)=6. \end{eqnarray*}$ Solution. $\overrightarrow{x}'=\left[\begin{array}{rr} 4&-3\\8&-6\end{array} \right]\overrightarrow{x},$ where $$\overrightarrow{x}(0)=\left[\begin{array}{c} 5\\6\end{array} \right] \mbox{ and } \overrightarrow{x}=\left[\begin{array}{c} x_1\\x_2\end{array} \right].$$

We can transform $$n$$th order linear ODE $$y^{(n)}=F(t,y,y',\ldots,y^{(n-1)})$$ to the following system of $$n$$ variables $$x_1=y,x_2=y',\ldots,x_n=y^{(n-1)}$$: $\begin{eqnarray*} x_1'&=& x_2\\ x_2'&=& x_3\\ &\vdots&\\ x_{n-1}'&=& x_n\\ x_n'&=& F(t,x_1,x_2,\ldots,x_n). \end{eqnarray*}$
Example. Convert the following third order linear ODE to a system of linear ODEs and write its matrix form. $y'''=e^ty''-ty'+\cos ty-5t.$ Solution. Let $$x_1=y,x_2=y',x_3=y''$$. Then $\begin{eqnarray*} x_1'&=& x_2\\ x_2'&=& x_3\\ x_3'&=& e^tx_3-tx_2+\cos tx_1-5t. \end{eqnarray*}$ Its matrix form is $\overrightarrow{x}'=\left[\begin{array}{rrr} 0&1&0\\0&0&1\\\cos t&-t&e^t\end{array} \right]\overrightarrow{x}+\left[\begin{array}{c} 0\\0\\-5t\end{array} \right],$ where $$\overrightarrow{x}=[x_1,\; x_2,\; x_3]^T$$.

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