Circuit Analysis and the Impulse Response

In this problem we explore the circuit response from an applied impulse voltage. Voltages being time dependent leads to a time analysis and the use of derivatives defined by the RC and LC circuit's respective constitutive laws. The first order non-linear differential equation defines the topology of the circuit and with no drive the function simplifies into the homogenous solution. This leads to our resistor voltage Vr and capacitor voltage Vc as a time dependent function. Giving an equation that can predict the outcome of the circuit at any time within a discrete range of time, as steady state in our perspective doesn't take long to reach.