Find the differential equation for Vo (RLC circuit)
Your equations are correct. Differentiate node \$\small V_2\$ equation and obtain an expression for \$\small V_1\$. Substitute this expression into the node \$\small V_1\$ equation. This gives a 2nd order equation in \$\small V_2\$.

Thus:

$$\small\ddot{V_2} +\left(\dfrac{1}{R_2C}+\dfrac{R_1}{L}\right)\dot{V_2}+\left(\dfrac{1}{LC}+\dfrac{R_1}{R_2 LC} \right)V_2=\dfrac{V_{in}}{LC}  $$