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: \[\begin{equation} \overrightarrow{x}'=A\overrightarrow{x}+\overrightarrow{g}, \;\;\;\;(50) \end{equation}\] 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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