While we solved the spring-mass system as a first order system of differential equations, it is in fact easier to find these solutions if we consider the whole equation as a single differential operator equation. This generalizes to all higher order linear differential equations with constant coefficients and we take a moment to explain this more elementary approach.
First, we make a definition.
As infinite dimensional vector spaces go, \(C^\infty (I, V)\) is not optimal and is often replaced with a Hilbert space. Nevertheless, for our purpose, it will do just fine. We will consider the case where \(V = \mathbb{C}\text{.}\) In this case, we have the linear transformation
\begin{equation*}
\frac{d}{dt} : C^\infty (I , \mathbb{C} ) \to C^\infty (I, \mathbb{C} ) .
\end{equation*}
This is the most basic of differential operators, but perhaps it is a new experience to consider this as a linear transformation. If we do, then we note that the eigenvalues are in fact all complex numbers! Indeed, taking \(\lambda \in \mathbb{C}\) we have the eigenfunction
\begin{equation*}
e^{\lambda t}
\end{equation*}
which spans the \(\lambda\)-eigenspace of the derivative. How can this be used? Well, let’s look at our homogeneous higher order differential equation again
\begin{equation*}
x^{(n)} (t) + a_{n - 1}x^{(n - 1)} (t) + \cdots + a_1 x^\prime (t) + a_0 x(t) = 0 .
\end{equation*}
and see that if the characteristic equation
\begin{equation*}
s^n + a_{n - 1} \lambda^{n - 1} + \cdots + a_1 s + a_0 = 0
\end{equation*}
has solutions \(\lambda_1, \ldots, \lambda_n\) then our equation becomes
\begin{equation*}
\left( \frac{d}{dt} - \lambda_1 I \right) \cdots \left( \frac{d}{dt} - \lambda_n I \right) x (t) = 0.
\end{equation*}
The expression
\begin{equation*}
\left( \frac{d}{dt} - \lambda_1 I \right) \cdots \left( \frac{d}{dt} - \lambda_n \right)
\end{equation*}
is a linear transformation from \(C^{\infty} (\mathbb{C})\) to itself and thus solutions to the homogeneous equation are just elements of the kernel of this transformation. If each of the \(\lambda_i\) are distinct, then it is not hard to see that these are just the eigenfunctions of the derivative
\begin{equation*}
e^{\lambda_1 t}, \ldots e^{\lambda_n t}.
\end{equation*}
In the case when there is multiplicity, we have a factor of the differential operator of the form
\begin{equation*}
\left( \frac{d}{dt} - \lambda I \right)^k .
\end{equation*}
Finding vectors in the kernel of this operator is precisely the same as finding generalized \(\lambda\)-eigenvectors of \(\frac{d}{dt}\text{.}\) In fact, the space of generalized \(\lambda\)-eigenvectors is
\begin{equation*}
e^{\lambda t}, t e^{\lambda t}, \frac{t^2}{2!} e^{\lambda t}, \ldots
\end{equation*}
so this derivative operator not only has every number as an eigenvalue, every eigenvalue has an infinite dimensional generalized eigenspace! Of course, only the first \(k\) generalized eigenvectors solve the equation
\begin{equation*}
\left( \frac{d}{dt} - \lambda I \right)^k x (t) = 0.
\end{equation*}
Thus if the characteristic polynomial is
\begin{equation*}
(s - \lambda_1)^{k_1} \cdots (s - \lambda_r)^{k_r}
\end{equation*}
then the general solution to the homogeneous differential equation (as a complex valued function) is
\begin{align*}
x (t) \amp = e^{\lambda_1 t} \left( C_{1,0} + C_{1,1} t + \cdots + C_{1, k_1 - 1} \frac{t^{k_1 - 1}}{(k_1 - 1)!} \right) + \cdots\\
\amp \cdots + e^{\lambda_r t} \left( C_{r,0} + C_{r,1} t + \cdots + C_{r, k_r - 1} \frac{t^{k_r - 1}}{(k_r - 1)!} \right)
\end{align*}
Now, the non-homogeneous case with eigenfunction \(f(t) = e^{\omega t}\) looks like
\begin{equation}
\left( \frac{d}{dt} - \lambda_1 I \right)^{k_1} \cdots \left( \frac{d}{dt} - \lambda_r I \right)^{k_r} x (t) = e^{\omega t} .\tag{5.4.8}
\end{equation}
If \(\omega\) is not one of the \(\lambda_i\text{,}\) one sees that
\begin{equation}
x_p (t) = \frac{e^{\omega t}}{(\omega - \lambda_1)^{k_1} \cdots (\omega - \lambda_r)^{k_r} }\tag{5.4.9}
\end{equation}
gives a solution. Using
Corollary 5.4.2 gives the general solution in this case.
