1. The problem is to express the differential equation $$y''' - y' + y = \cos t$$ in operator form. 2. Define the differential operator $$D = \frac{d}{dt}$$. 3. Then, $$y''' = D^3 y$$, $$y' = D y$$, and $$y = I y$$ where $$I$$ is the identity operator. 4. Substitute these into the equation: $$D^3 y - D y + I y = \cos t$$ 5. Factor out $$y$$: $$(D^3 - D + I) y = \cos t$$ --- 1. Express the differential equation $$y'' - 4y' + 4y = e^{2t}$$ in operator form. 2. Using $$D = \frac{d}{dt}$$, rewrite derivatives: $$y'' = D^2 y$$, $$y' = D y$$, $$y = I y$$. 3. Substitute: $$D^2 y - 4 D y + 4 I y = e^{2t}$$ 4. Factor out $$y$$: $$(D^2 - 4 D + 4 I) y = e^{2t}$$ --- 1. Express the differential equation $$y^{(4)} + y = 0$$ in operator form. 2. Using $$D = \frac{d}{dt}$$, rewrite derivatives: $$y^{(4)} = D^4 y$$, $$y = I y$$. 3. Substitute: $$D^4 y + I y = 0$$ 4. Factor out $$y$$: $$(D^4 + I) y = 0$$ --- 1. Express the differential equation $$y' + 2 y = e^{-t}$$ in operator form. 2. Using $$D = \frac{d}{dt}$$, rewrite derivatives: $$y' = D y$$, $$y = I y$$. 3. Substitute: $$D y + 2 I y = e^{-t}$$ 4. Factor out $$y$$: $$(D + 2 I) y = e^{-t}$$