Final answer:
To determine the form of a particular solution for the given differential equation, we can examine the right-hand side of the equation, which consists of two terms: t²e⁻⁸ᵗ and e⁻⁸ᵗ. We need to find a particular solution of the same form as the right-hand side.
Step-by-step explanation:
To determine the form of a particular solution for the given differential equation, we can examine the right-hand side of the equation, which consists of two terms: t²e⁻⁸ᵗ and e⁻⁸ᵗ. We need to find a particular solution of the same form as the right-hand side. Since the terms involve t and an exponential function, we can try a particular solution of the form yp(t) = At²e⁻⁸ᵗ + Be⁻⁸ᵗ. Here, A and B are arbitrary constants that we need to determine.