The theory for second-order linear non-homogeneous differential equations naturally extends to higher-order linear non-homogeneous differential equations.

Let’s consider the third-order linear constant-coefficient non-homogeneous differential equation \(y^{\,\prime \prime \prime }-y^{\,\prime }=1-2t\).

(a)
Let’s determine the characteristic polynomial for the complementary equation \(y^{\,\prime \prime \prime }-y^{\,\prime }=0\).
(b)
Let’s find the general solution to the complementary equation \(y^{\,\prime \prime \prime }-y^{\,\prime }=0\).
(c)
Let’s find a particular solution to \(y^{\,\prime \prime \prime }-y^{\,\prime }=1-2t\) by observing that if \(y\) is a quadratic polynomial then \(y^{\,\prime \prime \prime }=0\).
(d)
Now let’s write down the general solution to \(y^{\,\prime \prime \prime }-y^{\,\prime }=1-2t\).
(e)
Let’s solve the IVP \(y^{\,\prime \prime \prime }-y^{\,\prime }=1-2t\), \(y(0)=0\), \(y^{\,\prime }(0)=0\), \(y^{\,\prime \prime }(0)=0\).