Final answer:
To find the particular solution to the given differential equation, we identify a trial solution based on the nonhomogeneous part, substitute it into the equation, and solve for the unknown coefficients.
Step-by-step explanation:
The question asks for a particular solution to the second-order linear nonhomogeneous differential equation −3y′′ +4y′ −1y=1t² +1t−4e⁻⁴t. To find the particular solution, we first need to consider the form of the nonhomogeneous part (the right-hand side of the equation).
Polynomial terms like t² and t suggest a trial solution of the form At² + Bt + C. Meanwhile, the term e⁻⁴t suggests a trial solution of the form De⁻⁴t. As the right-hand side is a sum of these two types of terms, we attempt a solution that is a sum of these two forms: y_p = At² + Bt + C + De⁻⁴t.
Substituting y_p and its derivatives into the original equation, we'll end up with a system of equations for the coefficients A, B, C, and D. Once we solve this system, we will have determined the coefficients for the particular solution y_p.