Using the Method of Undetermined Coefficients, determine the form of a particular solution for the differential equation. (Do not evaluate coefficients.) \[ y^{\prime \prime}+81 y=6 t^{3} \sin 9 t \] The root(s) of the auxiliary equation associated with the given differential equation is/are (Use a comma to separate answers as needed.)

Step 1 ：The given differential equation is \(y^{\prime \prime}+81 y=6 t^{3} \sin 9 t\).

Step 2 ：The auxiliary equation of the given differential equation is obtained by replacing each derivative with a power of 'm'. So, for the given differential equation, the auxiliary equation would be \(m^2 + 81 = 0\).

Step 3 ：We need to solve this equation for 'm' to find the roots. The coefficients are a = 1, b = 0, c = 81.

Step 4 ：The discriminant D is calculated as \(D = b^2 - 4ac = -324\).

Step 5 ：The roots of the auxiliary equation are complex numbers, specifically ±9j. These roots indicate that the solution to the differential equation will involve sine and cosine functions.

Step 6 ：Final Answer: The roots of the auxiliary equation associated with the given differential equation are \(\boxed{-9j, 9j}\).