Solution:

To find the minimum value of n, we need to consider the properties of the given function y(x) = x sin(x) and analyze its behavior in relation to nth-order linear differential equations with constant coefficients.

Properties of y(x) = x sin(x):
- It is a product of two functions, x and sin(x).
- The function sin(x) is periodic with a period of 2π, meaning it repeats itself every 2π units.
- The function x is a linear function, increasing at a constant rate.

Analysis:
1. The function y(x) = x sin(x) is a solution of an nth-order linear differential equation with constant coefficients.
2. The presence of sin(x) suggests the differential equation may involve trigonometric functions.
3. The presence of x suggests the differential equation may involve derivatives of x.

Derivatives of y(x) = x sin(x):
To determine the differential equation satisfied by y(x), we need to find its derivatives.

- First derivative: y'(x) = x cos(x) + sin(x)
- Second derivative: y''(x) = -x sin(x) + 2cos(x)
- Third derivative: y'''(x) = -3cos(x) - xsin(x)
- Fourth derivative: y''''(x) = 4sin(x) - 3x cos(x)

Analysis of the derivatives:
- The first derivative involves both x and sin(x).
- The second derivative involves both x and sin(x), but with different coefficients.
- The third derivative involves both x and sin(x), but with different coefficients.
- The fourth derivative involves both x and sin(x), but with different coefficients.

Conclusion:
Based on the analysis of the derivatives, we can conclude that the given function y(x) = x sin(x) satisfies a 4th-order linear differential equation with constant coefficients. Therefore, the minimum value of n is 4, and the correct answer is option D).