Final answer:
To find a particular solution to the given differential equation using the Method of Undetermined Coefficients, you must guess a solution form adjusted for the nonhomogeneous term, then find its derivatives and equate coefficients to solve for the undetermined coefficients A and B.
Step-by-step explanation:
The student asked to find a particular solution to the differential equation x''(t) - 18x'(t) + 81x(t) = 4te9t using the Method of Undetermined Coefficients. To find the particular solution, we guess a solution of the form Ate9t since the nonhomogeneous term is a product of a polynomial and an exponential function. However, since e9t is a solution to the homogeneous equation, we must multiply our guess by t to account for the repeated root. Our guess then becomes At2e9t + Bte9t.
We then find the first and second derivatives of this guess:
- The first derivative: x'(t) = (2Ate9t + 9At2e9t + Be9t + 9Bte9t).
- The second derivative: x''(t) = (2Ae9t + 18Ate9t + 81At2e9t + 9Be9t + 81Bte9t).
Substitute x(t), x'(t), and x''(t) into the original differential equation and equate coefficients of like terms to find the values of A and B. Once these constants are determined, the particular solution can be formed.
It is left as an exercise to the student to prove that the obtained particular solution satisfies the differential equation by substituting the first and second derivatives with respect to time into the equation and verifying the identity.