What is the value of v₀(t) for the circuit shown in the figure, assuming zero initial conditions?

Given that, R=, L = 1H, C = 

The response for a series RLC circuit with source will be, V(t) = V_ss + (A₁ + A₂t)e^-at for a = W_0.

 a = = = 2

₀ =  = = 2.

a = ₀ ; system is critically damped.

At t = ; inductor is short circuit and capacitor is open circuit.

V(t) = Vss + (A1 + A2t)e-at

= 40 + A₁e^-2t + A₂te^-2t

At t = 0, voltage across capacitor is zero V(t) = 0.

 0 = 40 + A₁e^-2 + A₂te^-2

 A₁ + A₂ = -40

= 0 + -2 A₁e^-2t - 2A₂te^-2t + A₂e^-2t

0 = -2 (-40) - 0 + A₂

 A₂ = -80

Here, option 2 is correct.