Let’s consider the third order linear nonhomogeneous differential equation
\(y^{\,\prime \prime \prime }+3y^{\,\prime \prime }+3y^{\,\prime }+y=e^{-t}\).

(a)
Let’s determine the general solution for the complementary equation.
(b)
Is there a particular solution of the form \(y_p=Ae^{-t}\)?
(c)
Suppose we want a particular solution of the form \(y=ue^{-t}\). What must \(u\) satisfy? For reference, \(y^{\,\prime }=u^{\,\prime }e^{-t}-ue^{-t}\), \(y^{\,\prime \prime }=u^{\,\prime \prime }e^{-t}-2u^{\,\prime }e^{-t}+ue^{-t}\), and \(y^{\,\prime \prime \prime }=u^{\,\prime \prime \prime }e^{-t}-3u^{\,\prime \prime }e^{-t}+3u^{\,\prime }e^{-t}-ue^{-t}\).
(d)
Let’s determine the particular solution involving the least number of terms.
(e)
What’s the general solution?