Final answer:
To find the particular solution xₚ(t) for the differential equation x''(t) - 10x'(t) + 25x(t) = 4te⁵ᵗ, we assume a solution form and determine the coefficients by substituting into the original equation.
Step-by-step explanation:
The question involves finding a particular solution to a second-order linear nonhomogeneous differential equation using the Method of Undetermined Coefficients. The given differential equation is:
x''(t) - 10x'(t) + 25x(t) = 4te5t
The right-hand side of the equation, 4te5t, suggests that the particular solution, xp(t), will likely have the form Ate5t plus a polynomial term because the differential operator applied to the solution has terms that would generate a te5t from both the polynomial and exponential portions of the assumed solution.
The coefficients must be determined by substituting xp(t) and its derivatives into the original differential equation and solving for the coefficients. Once xp(t) is found, it contributes to the general solution of the differential equation together with the complementary solution.