$sx(s)=Ax(s)+Bu(s)$

$y(s)=Cx(s)+Du(s)$

Where $x(s)$ is a column vector containing the Laplace transforms of the $n$ state variables $x(s)=\left[\begin{array}{c}{x}_1(s)\\ \text{v}\text{d}\text{o}\text{t}\text{s}\\ x_n(s)\end{array}\right].$ The input is $U(s)$ and the output is $y(s).$ The output may or may not be a state variable, but it must be a linear combination of the state variables and of the input as shown in $(2).$ Equation $(1)$ is a system of coupled algebraic equations that are linear combinations of the state variables. The matrices $A,B,C,$ and $D$ contain the coefficients of the linear equations. The transfer function is given by Eq$.(3)$ below:

$T_p(s)=\frac{v_o(s)}{u(s)}=C[sI-A{]}^{-1}B+D$

For example, consider the following familiar low$-$pass filter that has been the subject of numerous analyses in lecture.

Defining the inductor current $i_L(s)$ and the capacitor voltage $v_C(s)$ as this system's two state variables and recognizing by inspection that $y(s)=v_o(s),$ write the A$,$ B$,$ C$,$ and D matrices for this system. You must derive first order differential equations for state variables by KVL and KCL$.$