Value of omega from forced response V0
By Ohm's Law $$v_o = IZ$$ where \$Z = Z_L + Z_C\$ is the combined impedance of the inductor and capacitor. You have a non-zero voltage source so you have a non-zero current \$I\$ through this impedance \$Z\$. In order to have \$v_o = 0\$ you therefore need $$Z = Z_L + Z_C = j\omega L + \frac{1}{j\omega C} = 0$$

If you simplify this equation you will arrive at $$\omega L = \frac{1}{\omega C}$$ as explained by Dave Tweed.