Homogenous solution is the correct answer.

Explanation:
When we assume the input x(n) of a system to be zero, it means that there is no external input or excitation to the system. In other words, the system is not being driven by any external source.

When the input to a system is zero, the output of the system is also zero. This is because the system does not have any input to process or respond to. Therefore, the output of the system in this case is also referred to as the homogenous or zero-input response.

The homogenous solution represents the inherent behavior of the system in the absence of any external input. It is determined solely by the system's characteristics, such as its transfer function or impulse response.

Example:
Consider a simple linear time-invariant (LTI) system described by the difference equation:

y(n) = 0.5*y(n-1) + x(n)

If we assume x(n) to be zero, the difference equation becomes:

y(n) = 0.5*y(n-1)

This is a recursive equation that describes the behavior of the system without any external input. Solving this equation will give us the homogenous solution, which represents the system's response when there is no input.

Importance:
The homogenous solution is important in understanding the behavior of a system and analyzing its stability. It helps in identifying the system's natural response and determining its stability properties.

In practical applications, the homogenous solution can be used to determine the system's response to initial conditions or transients. It provides insights into the system's dynamic behavior and can be used in conjunction with the particular solution (obtained when there is a non-zero input) to obtain the complete response of the system.

Therefore, assuming the input x(n) of a system to be zero helps in obtaining the homogenous solution, which represents the system's response in the absence of any external input.