Solving Current in an RLC Circuit

Given a series RLC circuit with the equation $L \frac{d^2 i}{dt^2} + R \frac{di}{dt} + \frac{1}{C} i = 0$ , where the inductance $L = 0.2$ Henrys, resistance $R = 330$ Ohms, and capacitance $C = 20$ microfarads, and initial conditions $i(0) = 0$ and $\frac{di}{dt}(0) = 30$ , solve the second-order differential equation for the current $i$ as a function of time $t$

In this problem, you are tasked with determining the function for current over time in a series RLC circuit, which is governed by a second-order differential equation. The circuit includes components of resistance, inductance, and capacitance, which together influence the rate of change of current. The problem presents initial conditions that are crucial for solving the differential equation accurately. This particular form of differential equation is common in analyzing circuits, oscillations, and waves, and is an excellent example of an engineering application of mathematics.

A series RLC circuit presents a classic example of a damped harmonic oscillator, where the interplay of resistive, inductive, and capacitive elements determines whether the system is underdamped, critically damped, or overdamped. Understanding the nature of the roots of the characteristic equation gives insights into the transient and steady-state behavior of the circuit. As you work through this problem, consider how the initial conditions impact the solution and how the characteristic polynomial relates to physical concepts inherent in electrical engineering.

The problem also highlights the utility of solving homogeneous linear differential equations and leveraging characteristic equations to find the natural response of electrical circuits. Through this problem, you further see the relevance of mathematical techniques like the Laplace Transform, although it isn't directly required here, as a powerful tool in simplifying complex circuit analysis into more manageable algebraic forms. This problem exemplifies the seamless integration of mathematical theory and practical electrical engineering applications.