Solution:

The given differential equation is (d^3 6d^2 12d 8)y=e^2x x^2 cos2x.

Finding the Characteristic Equation:
To find the characteristic equation, we assume that the solution of the differential equation is in the form of y = e^(rt). Then, we substitute it in the given differential equation.

(d^3 6d^2 12d 8)y = (r^3 6r^2 12r 8)e^(rt)
(e^2x x^2 cos2x) = e^(2x) (x^2 cos2x)

After substitution, we get:

(r^3 6r^2 12r 8)e^(rt) = e^(2x) (x^2 cos2x)

We can simplify this equation by dividing both sides by e^(rt).

r^3 6r^2 12r 8 = e^(2x) (x^2 cos2x) / e^(rt)

r^3 6r^2 12r 8 = x^2 cos2x e^(2x-rt)

Next, we solve for r by finding the roots of the characteristic equation.

r^3 6r^2 12r 8 = 0

(r+2)(r+2)(r+2) = 0

r = -2 (multiplicity 3)

Therefore, the characteristic equation is (r+2)^3 = 0.

Finding the General Solution:
Since the characteristic equation has a triple root, the general solution can be expressed as:

y = (c1 + c2x + c3x^2)e^(-2x) + c4xe^(-2x)sin2x + c5xe^(-2x)cos2x

where c1, c2, c3, c4, and c5 are constants to be determined.

Finding the Particular Solution:
To find the particular solution, we use the method of undetermined coefficients. Since the right-hand side of the differential equation is e^(2x) x^2 cos2x, we assume that the particular solution is of the form:

y = (a + bx + cx^2) e^(2x) cos2x + (d + ex + fx^2) e^(2x) sin2x

where a, b, c, d, e, and f are constants to be determined.

Differentiating this equation three times and substituting it in the differential equation, we get:

12a + 24b + 12c = 0
-24a - 48b - 24c = 0
-48a - 96b - 48c + 16d + 32e + 16f = x^2 cos2x e^(2x)
-48a - 96b - 48c - 32d - 64e - 32f = 0
48a + 96b + 48c - 16d - 32e - 16f = 0
48a + 96b + 48c + 32d + 64e + 32f = 0

Solving these equations