The way this is generally done is to get the equations into a matrix form (or solve them.
\$\begin{bmatrix} \dot{x} \end{bmatrix}=
\begin{bmatrix} Ax \end{bmatrix}+ \begin{bmatrix}Bf\end{bmatrix}\$
or this notation since the state would be \$\dot{x}=\frac{di}{dt}\$
\$\begin{bmatrix} \frac{di}{dt} \end{bmatrix}=
\begin{bmatrix} Ax \end{bmatrix}+ \begin{bmatrix}Bf\end{bmatrix}\$
The matrix coefficients are the same for fly speck or differential notation.
$$
2\frac{di_{1}}{dt} - 2\frac{di_{2}}{dt} + 0\frac{di_{3}}{dt} = -i_{1} + 0i_{2} +0i_{3} +v_{i}(t)
$$
$$
2\frac{di_{1}}{dt} - 3\frac{di_{2}}{dt} + \frac{di_{3}}{dt} = 0i_{1}+ 3i_{2}+0i_{3}
$$
$$
0\frac{di_{1}}{dt} + \frac{di_{2}}{dt} - \frac{di_{3}}{dt} = 0i_{1}+0i_{2} + i_{3} + v_{o}(t)
$$
You can represent a matrix form from the above equations:
\$\begin{bmatrix} 2 & -2 & 0 \\ 2&-3&1 \\ 0&1&-1 \end{bmatrix}=
\begin{bmatrix} -1& 0& 0 \\0&3&0 \\0 &0&1 \end{bmatrix}+ \begin{bmatrix} v_{i}(t) \\ 0\\v_{o}(t) \end{bmatrix}\$
You need to reduce the system of equations until you get this (all the differential equations are reduced, this form makes it easy to work with or use it for simulation)
\$\begin{bmatrix} 1 & 0 & 0 \\ 0&1&0 \\ 0&0&1 \end{bmatrix}=
\begin{bmatrix} Blah \end{bmatrix}+ \begin{bmatrix}vin blah\end{bmatrix}\$
if you substitued the variables back in it would look like this:
\$\begin{bmatrix} \frac{di_1}{dt} & 0 & 0 \\ 0&\frac{di_2}{dt}&0 \\ 0&0&\frac{di_3}{dt} \end{bmatrix}=
\begin{bmatrix} Blah \end{bmatrix}+ \begin{bmatrix}vin blah\end{bmatrix}\$
