In order to write the equations in the state-variable form (i.e., a set of simultaneous first-order differential equations), we select the capacitor voltages v_1 \text{ and } v_2 as the state elements (i.e., x = [v_1 v_2 ]^T ) and v_i as the input (i.e., u = v_i ). Here v_1 = v_2, v_2 = v_1 − v_3, and still v_1 = v_i. Thus v_1 = v_i, v_2 = v_1 , and v_3 = v_i − v_2 . In terms of v_1 and v_2 , Eq. (2.34) is

\frac{v_1 − v_i}{R_1} + \frac{v_1 − (v_i − v_2 )}{R_2} + C_1 \frac{dv_1}{dt} = 0.

-\frac{v_{1}-v_{2}}{R_{1}}+\frac{v_{2}-v_{3}}{R_{2}}+C_{1} \frac{d v_{2}}{d t}=0 ,           (2.34)

Rearranging this equation into standard form, we get

\frac{dv_1}{dt} = − \frac{1}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) v_1 − \frac{1}{C_1} \left( \frac{1}{R_2} \right) v_2 + \frac{1}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right) v_i.                   (7.6)

In terms of v_1 and v_2 , Eq. (2.35) is

\frac{v_i − v_2 − v_1}{R_2} + C_2 \frac{d}{dt}(v_i − v_2 − v_i) = 0.

\frac{v_{3}-v_{2}}{R_{2}}+C_{2} \frac{d\left(v_{3}-v_{1}\right)}{d t}=0               (2.35)

In standard form, the equation is

\frac{dv_2}{dt} = − \frac{v_1}{C_2R_2} − \frac{v_2}{C_2R_2} + \frac{v_i}{C_2R_2}.             (7.7)

Equations (2.34)–(2.35) are entirely equivalent to the state-variable form, Eqs. (7.6) and (7.7), in describing the circuit. The standard matrix definitions are

\pmb F = \left[\begin{matrix} -\frac{1}{C_1}\left(\frac{1}{R_1}+ \frac{1}{R_2} \right) & -\frac{1}{C_1} \left( \frac{1}{R_2}\right)\\ -\frac{1}{C_2R_2} & -\frac{1}{C_2R_2} \end{matrix} \right] , \\ \pmb G = \left[\begin{matrix} \frac{1}{C_1}\left( \frac{1}{R_1} + \frac{1}{R_2}\right) \\ \frac{1}{C_2R_2} \end{matrix} \right] , \\ \pmb H = \left[\begin{matrix} 0 & -1 \end{matrix} \right] , \ \ \ J =1 .