Finding the state space representation of a circuit
The current going through \$C_2\$ and \$C_3\$ is the same, so, by,
$$ Q = Cv \leftrightarrow  i=C\dot{v}$$
For that branch,

$$\frac{\dot{v_3} - \dot{v_2} }{C_2} = \frac{-\dot{v_3}}{C_3}.$$

Also, the current that goes through the resistor either goes through \$C_1\$ or the branch with \$C_2, C_3\$. By associating them in series and then parallel, we find that,

$$\frac{v_1 - v_2 }{R} = \frac{ \dot{v_2}}{C_1+(C_2^{-1}+C_3^{-1})^{-1}}.$$

I did not go through the whole process of figuring out the state equations, but these should show that \$\dot{v_3} = k \dot{v_2}\$, those derivatives are related by a constant.