Final answer:
A suitable form for Y(t) using the method of undetermined coefficients is given by Y(t) = Jte⁹ᵗ + Kte⁻⁷ᵗ + Le⁻ᵗsint + Me⁻ᵗcost + Q.
Step-by-step explanation:
To determine a suitable form for Y(t) using the method of undetermined coefficients, we need to consider that the given equation is a linear differential equation of the form y⁴ + 2y′′′ + 2y ′′ = 8e⁹ᵗ + 8te⁻⁷ᵗ + e⁻ᵗsint. We can see that this equation involves exponential terms and a sine function. So, a suitable form for Y(t) would be a linear combination of the form Y(t) = Jte⁹ᵗ + Kte⁻⁷ᵗ + Le⁻ᵗsint + Me⁻ᵗcost + Q, where J, K, L, M, and Q are coefficients to be determined.