Final answer:
The resonance part of the solution usually represents sinusoidal oscillation and occurs when the frequency matches the system's natural resonant frequencies, which are integer multiples of the fundamental frequency. This is associated with large amplitude oscillations and is significant in both mechanical and electrical resonant systems.
Step-by-step explanation:
The resonance part of the solution that represents a system's response without taking the complementary solution into account typically involves sinusoidal oscillation. In the context of an oscillating system, such as a guitar string or a circuit, the resonant frequencies are those at which the system naturally tends to oscillate.
These frequencies are integer multiples of the fundamental frequency, leading to harmonics and overtone patterns. At these resonant frequencies, a system can exhibit free vibration without a driving force. In circuits, the resonant frequency is where the impedance is minimized and is calculated based on the values of inductance (L) and capacitance (C).
When the frequency matches one of these natural frequencies, resonance occurs, often resulting in large amplitude oscillations. Distinguishing characteristics of resonance include harmonics, overtones, and standing waves as described for sound within a tube near a tuning fork. Oscillations at these frequencies are not damped or exponential, and they differ from the motion described by imaginary or real roots.
Therefore answer is d) Sinusoidal oscillation.