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

We determine the characteristic polynomial/equation by assuming that there are solutions of the form \(y=e^{rt}\). For a factor of the form \((r-a)^k\) in the characteristic polynomial we get a contribution of \((c_1+c_2 t+\cdots +c_{k}t^{k-1})e^{at}\) in the general solution and for a non-repeated quadratic factor in the characteristic polynomial that has roots at \(\lambda \pm \omega i\) we get a contribution of
\(e^{\lambda t}(c_1\cos {(\omega t)}+c_2\sin {(\omega t)})\) in the general solution.

Let’s consider the third-order linear constant-coefficient homogeneous differential equation \(y^{\,\prime \prime \prime }-y^{\,\prime \prime }-y^{\,\prime }+y=0\).

(a)
Let’s determine the characteristic polynomial (assuming there are solutions of the form \(y=e^{rt}\)).
(b)
Let’s factor this as much as possible and determine the roots (aka zeros).
(c)
Let’s find the general solution to \(y^{\,\prime \prime \prime }-y^{\,\prime \prime }-y^{\,\prime }+y=0\).
(d)
Let’s solve the IVP \(y^{\,\prime \prime \prime }-y^{\,\prime \prime }-y^{\,\prime }+y=0\), \(y(0)=1\), \(y^{\,\prime }(0)=0\), \(y^{\,\prime \prime }(0)=3\).
(e)
How would the general solution change if the differential equation were \(y^{\,\prime \prime \prime \prime }-y^{\,\prime \prime \prime }-y^{\,\prime \prime }+y^{\,\prime }=0\)?

Let’s consider the fourth-order linear constant-coefficient homogeneous differential equation \(y^{\,(4)}-y=0\).

(a)
What’s the characteristic polynomial? What roots does it have?
(b)
What’s the general solution?