Understanding RLC Circuits: Critical Damping & Homework Equations

[dwn]

Homework Statement

Image

Homework Equations

Critical Damping

* related to '*' on image : di_L/dt = -v_c(0) - Ri(0)
When do we use this to solve for the other constant? In the text, an example problem is never fully explained in mathematical terms.

The Attempt at a Solution

I am looking for someone to kind of explain what to keep an eye out for when analyzing RLC circuits, or provide a source. I have looked through a couple different textbooks and certain things vary and its leading to confusion. One particular concern is after we calculate whether a circuit is over/under damped or critical (I know the characteristic equations for each), are we solving v_c or i_L. Now I'm sure it depends on what the question asks, but sometimes it includes both instances. Can someone please sort this out for me?**EDIT**
The voltage for capacitor should be 12/5 = 2.4 Correct?

 

[rude man]

**EDIT**
The voltage for capacitor should be 12/5 = 2.4 Correct?

Incorrect. What is the total resistance in series with L?

Also, your t(0+) diagram is wrong. The 1 ohm resistor has mysteriously migrated away from C!