Final answer:
To find a particular solution to the given differential equation, we use the forms of the non-homogeneous terms (polynomial and exponential). By substituting these particular solutions into the equation and solving for the coefficients, we can find the particular solution.
Step-by-step explanation:
The given differential equation is -2y'' - y' + y = t² - 2t + 2e⁻³ᵗ
To find a particular solution to this equation, we need to consider the form of the non-homogeneous terms. In this case, it consists of a polynomial (t² - 2t) and an exponential term (2e⁻³ᵗ).
For the polynomial term, we can assume a particular solution of the form yp = At² + Bt + C. And for the exponential term, we can assume a particular solution of the form yp = De⁻³ᵗ.
Substituting these particular solutions into the differential equation and solving for the coefficients A, B, C, and D will give us the particular solution to the equation.