Final answer:
To find a particular solution, x_p(t), to the differential equation x''(t) - 10x'(t) + 25x(t) = 132t²e⁵ᵗ, we must guess a solution that matches the form of the right side of the equation and then verify it by substituting the solution and its derivatives back into the equation.
Step-by-step explanation:
The student's question is about finding a particular solution to the differential equation x''(t) - 10x'(t) + 25x(t) = 132t²e⁵ᵗ using the method of undetermined coefficients. To determine the particular solution, xp(t), we need to make an educated guess of the form of the particular solution based on the right side of the equation. Since the right side is a product of a polynomial and an exponential function, the guess for xp(t) should have the same form. After finding the appropriate form of xp(t), we differentiate it twice to get x''p(t) and x'p(t), and then substitute these along with xp(t) back into the differential equation to determine the coefficients of the guessed solution.
It is an exercise to show that the guessed solution meets the original differential equation. This process involves taking the guessed xp(t), its first derivative x'p(t), and second derivative x''p(t), plugging them into the left side of the equation and verifying that it equals the right side, 132t²e⁵ᵗ. If they are equal, xp(t) is indeed a particular solution to the differential equation.