Undetermined coefficients naturally applies to higher-order linear non-homogeneous differential equations. Let’s consider the third-order linear constant-coefficient non-homogeneous differential equation \(y^{\,\prime \prime \prime }-2y^{\,\prime \prime }-4y^{\,\prime }+8y=\dfrac {27t}{e^t}\).

(a)
Let’s determine the characteristic polynomial for the complementary equation \(y^{\,\prime \prime \prime }-2y^{\,\prime \prime }-4y^{\,\prime }+8y=0\).
(b)
Let’s find the general solution to the complementary equation
\(y^{\,\prime \prime \prime }-2y^{\,\prime \prime }-4y^{\,\prime }+8y=0\).
(c)
Let’s find a particular solution to \(y^{\,\prime \prime \prime }-2y^{\,\prime \prime }-4y^{\,\prime }+8y=\dfrac {27t}{e^t}\) by using that if \(y\) is a linear polynomial multiple of \(e^{-t}\) then it and its derivatives (of any order) are linear or constant polynomial multiples of \(e^{-t}\).
(d)
Now let’s write down the general solution to \(y^{\,\prime \prime \prime }-2y^{\,\prime \prime }-4y^{\,\prime }+8y=\dfrac {27t}{e^t}\).
(e)
Let’s solve the IVP \(y^{\,\prime \prime \prime }-2y^{\,\prime \prime }-4y^{\,\prime }+8y=\dfrac {27t}{e^t}\), \(y(0)=-1\), \(y^{\,\prime }(0)=5\), \(y^{\,\prime \prime }(0)=-3\).