Find state space representation with 2 inputs and 2 outputs
A possible solution: 



Choosing state variables as current in the inductor and the voltage in the capacitor:

$$ x_1 = i_L = i_1 $$ 
$$x_2 = v_C $$

Applying KVL and KCL to the left mesh and top node, respectively: 

$$ v_1 - R_1x_1 -L\dot{x_1}-x_2=0 $$
$$ -x_1 + C\dot{x_2} + \frac{x_2-v_2}{R_2} = 0 $$

Rearranging:

$$ \dot{x_1}= -\frac{R_1}{L}x_1 -\frac{1}{L}x_2 + \frac{1}{L}v_1$$
$$ \dot{x_2}=\frac{1}{C}x_1-\frac{1}{CR_2}x_2 + \frac{1}{CR_2}v_2 $$

In standard state space representation:
$$ \mathbf{\dot{x}=Ax+Bu} $$
$$ \mathbf{y = Cx+Du}$$

And, since that \$i_2=\frac{x_2-v_2}{R_2}\$:

$$\begin{bmatrix}\dot{i_1}\\\dot{v_c}\end{bmatrix}=\begin{bmatrix}-R_1/L&amp;-1/L\\1/C&amp;-1/CR_2\end{bmatrix}\begin{bmatrix}i_1\\v_c\end{bmatrix}+\begin{bmatrix}1/L&amp;0\\0&amp;1/CR_2\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}$$

$$\begin{bmatrix}i_1\\i_2\end{bmatrix}=\begin{bmatrix}1&amp;0\\0&amp;1/R_2\end{bmatrix}\begin{bmatrix}i_1\\v_c\end{bmatrix}+\begin{bmatrix}0&amp;0\\0&amp;-1/R_2\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}$$