Final answer:
To find a particular solution to the given differential equation using the Method of Undetermined Coefficients, we assume the particular solution to be of the form y_p(t) = Ae^(2t)cos(9t) + Be^(2t)sin(9t), where A and B are undetermined coefficients. We then substitute this particular solution and its derivatives into the original differential equation to solve for the coefficients and obtain the complete particular solution.
Step-by-step explanation:
To find a particular solution to the given differential equation using the Method of Undetermined Coefficients, we first need to determine the form of the particular solution. The given equation can be written as: y'' + 8y' + 16y = 1521e^2tcos(9t). The right-hand side of the equation represents a combination of exponential and trigonometric functions, so the particular solution will have the same form. We assume the particular solution to be of the form: y_p(t) = Ae^(2t)cos(9t) + Be^(2t)sin(9t), where A and B are undetermined coefficients.
Taking the first and second derivatives of y_p(t), we get: y'_p(t) = (2Ae^(2t) - 9Be^(2t)sin(9t)) and y''_p(t) = (4Ae^(2t) - 36Be^(2t)cos(9t)).
Now, substitute the particular solution and its derivatives into the original differential equation and equate the coefficients of similar terms. The constant terms must be equal, the coefficients of e^(2t) terms must be equal, and the coefficients of the trigonometric terms must be equal.
Solving the resulting system of equations will give us the values of A and B. Finally, substituting the values of A and B back into the assumed form of the particular solution, we obtain the complete particular solution: y_p(t) = 3e^(2t)cos(9t) + 2e^(2t)sin(9t).