Consider the ODE

$y^{\text{'}\text{'}}(t)-8y^{\text{'}}(t)16y(t)=R(t)$

$($a$)$ The general solution $y_H(t)$ of the corresponding homogenous ODE is

$y_H(t)=$

$$

Use A and B as your arbitrary constants.

$($b$)$ Suppose that $R(t)=3sin(2t).$ Which of these would be an appropriate form to try for the particular solution $y_P(t)?$

$3sin(2t)$

asin$(2t)$

$3sin(at)$

$sin(at)cos(bt)$

asin$(2t)$ acos$(2t)$

asin$(2t)$ bcos$(2t)$

atsin$(2t)$

$($c$)$ Suppose that $R(t)=3e^{2t}.$ Which of these would be an appropriate form to try for the particular solution $y_P(t)?$

$3e^{2t}$

$3e^{at}$

$a{e}^{2t}$

$e^{at}$

$t{e}^{at}$

$at{e}^{2t}$

$t^2{e}^{at}$