The equation
\begin{equation*}
m x'' + bx' + kx = g(t)
\end{equation*}
can be used to model a harmonic oscillator where forcing occurs. In general, we will not be able to solve this equation explicitly for a given \(g(t)\text{;}\) however, certain forcing functions often occur in practice. Some of the more important forcing functions are \(g(t) = e^{-at}\text{,}\) where the external force decreases exponentially over time; \(g(t) = k\text{,}\) where a constant force is applied; and \(g(t) = \cos \omega t\) or \(g(t) = \sin \omega t\text{,}\) where a force is applied periodically.
In the case of the unforced damped harmonic oscillator,
\begin{equation*}
m x'' + b x' + kx =0,
\end{equation*}
we know that \(m \gt 0\text{,}\) \(b \gt 0\text{,}\) and \(k \gt 0\text{.}\) Thus, we can rewrite this equation as
\begin{equation*}
x'' + px' + q x = 0,
\end{equation*}
where \(p = b/m\) and \(q = k/m\) are both positive. The characteristic equation is \(\lambda^{2} + p\lambda + q = 0\text{,}\) and the roots are
\begin{equation*}
\lambda_{1} = \frac{-p + \sqrt{p^{2}-4q}}{2} \quad \text{and} \quad \lambda_{2} = \frac{-p - \sqrt{p^{2}-4q}}{2}.
\end{equation*}
As we know from
Theorem 2.1.10, there are three distinct possibilities for the general solution, corresponding to whether
\(p^{2}-4q\) is positive, zero, or negative. Explicitly, general solutions for the undamped harmonic oscillator are given by
\begin{equation*}
x(t)= \begin{cases}
c_{1}e^{\lambda_{1} t} + c_{2}e^{\lambda_{2} t} \quad & p^{2}-4q \gt 0 \\
c_{1}e^{\alpha t} + c_{2}te^{\alpha t} \quad & p^{2}-4q =0 \\
c_{1}e^{\alpha t}\cos(\beta t) + c_{2}e^{\alpha t}\sin(\beta t) \quad & p^{2} - 4q \lt 0
\end{cases}
\end{equation*}
where \(\alpha = -\frac{p}{2}\) and \(\beta = \frac{\sqrt{4q - p^{2}}}{2}\text{.}\) Since \(p \gt 0\text{,}\) \(\alpha \lt 0\text{,}\) and thus for any constants \(c_{1}\) and \(c_{2}\) we have
\begin{equation*}
\lim_{t \to \infty}(c_{1}e^{\alpha t} + c_{2}te^{\alpha t}) = \lim_{t \to \infty}(c_{1}e^{\alpha t}\cos(\beta t) + c_{2}e^{\alpha t}\sin(\beta t)) = 0.
\end{equation*}
Moreover, both \(\lambda_{1}\) and \(\lambda_{2}\) are strictly negative when \(p^{2}-4q \gt 0\text{.}\) (To see that this holds for \(\lambda_{1}\text{,}\) we must also note that \(q \gt 0\) in this setting.) So
\begin{equation*}
\lim_{t \to \infty}(c_{1}e^{\lambda_{1} t} + c_{2}e^{\lambda_{2} t}) = 0
\end{equation*}
as well, for any constants \(c_{1}\) and \(c_{2}\text{.}\)
Theorem 2.2.3.
Suppose
\begin{equation*}
x'' + px' + q x = g(t)
\end{equation*}
has solution \(x = x_h + x_p\text{,}\) where \(p \gt 0\) and \(q \gt 0\text{.}\) If
\begin{equation*}
x = x_h + x_p
\end{equation*}
is the general solution to the equation, then
\begin{equation*}
x = x_h + x_p \to x_p
\end{equation*}
as \(t \to \infty\text{.}\)
In other words, all solutions of a damped harmonic oscillator with nonzero damping are essentially the same for large values of \(t\text{.}\)
