Final answer:
To solve the given differential equation using the method of undetermined coefficients, assume a particular solution in the form that matches the right-hand side of the equation and find the undetermined coefficients by substituting the assumed solution into the equation.
Step-by-step explanation:
To find a solution to the non-homogeneous second-order linear differential equation with constant coefficients using the method of undetermined coefficients, we'll look at the right-hand side of the equation y'' - 8 y' + 37 y = 16e⁴ᵗ cos(5t) + 96e⁴ᵗsin (5t) + 2e²ᵗ. The right-hand side consists of terms that are products of exponentials and trigonometric functions, which suggest the particular solution, yₚ, will have a similar form.
To construct yₚ, we assume a solution of the form:
yₚ = e⁴ᵗ(A cos(5t) + B sin(5t)) + Ce²ᵗ
Where A, B, and C are the undetermined coefficients we aim to find. We differentiate yₚ twice and substitute yₚ, yₚ', and yₚ'' into the original differential equation to find values for these coefficients. Once we have the coefficients, we will have found the particular solution to the differential equation.
Remember, the complete solution y(t) to the differential equation would be y(t) = yₛ(t) + yₚ, where yₛ(t) represents the complementary solution involving the homogeneous part of the differential equation.