State variable equations for a RLC circuit
The inductor equation:
$$L i_3'=v_1-v_C$$

The capacitor equation:
$$c v_c'=i_3\ \ \ (1)$$

Kirchoff's current law:
$$\frac{u-v_1}{R_1}=i_3+\frac{v_1}{R_2}$$

which can be solved for \$v_1\$ to get
$$v_1=\frac{R_2 \left(u-i_3 R_1\right)}{R_1+R_2}$$

Substitute this in the inductor equation 
$$L i_3'=\frac{R_2 \left(u-i_3 R_1\right)}{R_1+R_2}-v_c \ (2)$$

The equations (2) and (1) can then be put in matrix form as

$$\left(
\begin{array}{c}
 i_3' \\
 v_c' \\
\end{array}
\right)=\left(
\begin{array}{cc}
 -\frac{R_1 R_2}{L \left(R_1+R_2\right)} &amp; -\frac{1}{L} \\
 \frac{1}{c} &amp; 0 \\
\end{array}
\right).\left(
\begin{array}{c}
 i_3 \\
 v_c \\
\end{array}
\right)+
\left(
\begin{array}{c}
 \frac{R_2}{L \left(R_1+R_2\right)} \\
 0 \\
\end{array}
\right) u$$

Verifying using Mathematica.