Electrical – Calculating the damping ratio, \$ \zeta \$, and the undamped natural frequency, \$ \omega_0\$

My simple analysis

Since V(L)=LdI/di = "Vx" is in series with A*Vx for A=4 , then R damping voltage sees (1+4=5) or 5*L as effective inductance.

Since Ic=CdV/dt = "Iy" boosted by B*Iy, for B= 3, then the circuit produces (3+1 = 4x) or 4 effective capacitance multiplier.

So the \$L'=5L,~ C'=4C,~~~~~ L'C'=10^{-4}= 5*2.5mH*4*2mF\$

I know that I can apply here \$\omega _0=\dfrac{1}{\sqrt{L'C'}}=100 ~rad/s \$

Z(L') @ ωo =2π100 rad/s 2.5mH = 1.571 Ω at resonance

R=50Ω is higher than Z(ωoL) in parallel with the current source so underdamped <<1

Damping Coefficient = ζ =0.0125=α/ωo=1.25/100 using your value for α