Chapter 4.3, Problem 16P

a.

To determine

To show: $D-2\text{ and }{\left(D+1\right)}^2$ are the annihilators of the terms on the RHS of the given equation.

b.

To determine

To show: The equation ${\left(D-2\right)}^4{\left(D+1\right)}^{3}Y=0$ is obtained on applying the operator $\left(D-2\right){\left(D+1\right)}^2$ to the given equation and hence show that, $Y\left(t\right)=c_1{e}^{2t}+c_{2}t{e}^{2t}+c_3{t}^2{e}^{2t}+c_4{t}^3{e}^{2t}+c_5{e}^{-t}+c_{6}t{e}^{-t}+c_7{t}^2{e}^{-t}$ is a solution of the equation.

c.

To determine

The particular solution of the given equation.