So we were asked to simulate this circuit over at LT SpiceIV:

Tapping the voltage probe above the capacitor and tapping the current probe on the capacitor will give the following graph:

(V(n003) means the voltage at node 3)

Analysis would be easy as this is a second order circuit, but we (me and my classmates) were rather puzzled as these were the questions that came after the instructed simulation:

1. Generate the 1st order ordinary linear differential equations (ODE) for Fig. 1. Solve for V(n003) the voltage across capacitor.

2. Transform your 1st order system of linear ODEs into Laplace Transform and solve for L{Vn003)}. Then solve for L-1(Vn003) to get V(n003).

3. Compare your results in item 1 with item 2.

4. Derive the current across the capacitor.

5. Using Scilab, for t=[0:.5:20], plot V(n003). Compare your result with Fig 3 V(n003).

6. Using Scilab, for t=[0:.5:20], plot I(C1). Compare your result with Fig 3 I(C1).

2. Transform your 1st order system of linear ODEs into Laplace Transform and solve for L{Vn003)}. Then solve for L-1(Vn003) to get V(n003).

3. Compare your results in item 1 with item 2.

4. Derive the current across the capacitor.

5. Using Scilab, for t=[0:.5:20], plot V(n003). Compare your result with Fig 3 V(n003).

6. Using Scilab, for t=[0:.5:20], plot I(C1). Compare your result with Fig 3 I(C1).

You might be surprised as well because we were asked for a first order ODE, when the circuit clearly is a second order ODE. The second order ODE can be obtained by using KVL around the circuit, giving:

4i + Li' + i/C integral(i dt) = 9

and differentiating every term to remove the integral gives:

Li'' + 4i' + i/C = 0

substituting values from the circuit:

i'' + 4i' + 4i = 0

Solving for the natural response and the forced response (which will be useless since r(x) is 0) would then give:

i(t) = Ae^-2t + Bte^-2t

If it was the usual ODE question, values of i(0) and i'(0) would be given in order to solve for constants A and B. But in this setup, you'll only manage to get i(0) and i'(0) will be hard to find. Now you may ask why am I solving for the current when number 1 asks for the voltage across the capacitor - since the current is constant (series circuit), I can solve for it then simply plug the equation of the constant into the voltage equation for a capacitor then there I have it.

I guess what our professor wants is an ODE that would give the curve for the voltage (duh)...

Or did I get things wrong guys? Because number 2 asks us to solve for the same 1st order ODE using Laplace. If I use Laplace on the second order ODE, I'd have problems right away since the transform of i'' would ask for the value of i'(0).

If anyone has a clue on how I'm supposed to get a first order ODE to get the voltage please do help. I'm feeling really stupid here because I'm starting to get the feeling that this is supposed to be real easy.

P.S. the DC Steady State would also work fine, it will give the voltage across the capacitor at t = 0- and t = 0+ but it won't give the curve we want.