Given:
The particular solution is given as y_p(x) = x * cos(2x).
The differential equation is given as f(y^n) + ay = -4sin(2x).

To find:
The value of the constant a.

Solution:

Step 1: Find the nth derivative of y_p(x)
The given particular solution is y_p(x) = x * cos(2x).
To find the nth derivative, we use the product rule and chain rule repeatedly.

The first derivative of y_p(x) with respect to x is:
y_p'(x) = cos(2x) - 2x * sin(2x)

The second derivative of y_p(x) with respect to x is:
y_p''(x) = -4sin(2x) - 4x * cos(2x)

The third derivative of y_p(x) with respect to x is:
y_p'''(x) = -12cos(2x) + 8x * sin(2x)

The fourth derivative of y_p(x) with respect to x is:
y_p''''(x) = 32sin(2x) + 16x * cos(2x)

We can observe a pattern in the nth derivatives of y_p(x):
- The odd derivatives have a coefficient of (-1)^(n/2) * (2n)!
- The even derivatives have a coefficient of (-1)^(n/2) * (2n)!

Step 2: Substitute the particular solution and its nth derivative into the differential equation
Substituting y_p(x) = x * cos(2x) and its fourth derivative into the given differential equation, we have:
f(y^4) + ay = -4sin(2x)

The fourth derivative of y_p(x) is given as:
y_p''''(x) = 32sin(2x) + 16x * cos(2x)

Substituting these values into the differential equation, we get:
32sin(2x) + 16x * cos(2x) + a(x * cos(2x)) = -4sin(2x)

Step 3: Simplify the equation and find the value of a
Simplifying the equation, we have:
32sin(2x) + 16x * cos(2x) + ax * cos(2x) = -4sin(2x)

Grouping the terms with sin(2x) and cos(2x), we get:
(32 + a)sin(2x) + (16x + ax)cos(2x) = 0

Since sin(2x) and cos(2x) are linearly independent, the coefficients of both terms must be zero:
32 + a = 0 (Equation 1)
16x + ax = 0 (Equation 2)

From Equation 2, we can see that either x = 0 or a + 16 = 0.

If x = 0, then y_p(x) = 0 * cos(2 * 0) = 0, which is not a valid solution.

Therefore, a + 16 = 0, which implies a = -16.