A second-order differential equation characterizes a source-free series RLC circuit, marking its distinct mathematical representation. The complete solution of this equation is a blend of two unique solutions, each linked to the circuit's roots expressed in terms of the damping factor and resonant frequency.

When the damping factor surpasses the resonant frequency, both roots are real and negative, leading to an overdamped response. In this scenario, the circuit's reaction gradually decays over time.

When the damping factor matches the resonant frequency, the second-order differential equation simplifies to a first-order equation with an exponential solution. The natural response follows a pattern of peaking at its time constant and then decaying to zero, signifying critical damping.

For situations where the damping factor is less than the resonant frequency, complex roots emerge, characterized by the damped natural frequency. Euler's formula simplifies the complete response to sine and cosine functions, resulting in an underdamped and oscillatory natural response with a time period proportional to the damped natural frequency.

These different response behaviors illustrate the significance of source-free RLC circuits in circuit analysis, offering intriguing insights into electrical circuit behavior and applications.