functions that are periodic are especially important. Recall that a function \(g(t)\) is periodic if

\begin{equation*}
g(t + T) = g(t)
\end{equation*}

for all \(t\) and some fixed constant \(T\text{.}\) The most familiar periodic functions are

\begin{equation*}
g(t) = \sin \omega t \mbox{ and } g(t) = \cos \omega t.
\end{equation*}

The period for each of these two functions is \(2 \pi / \omega\) and the frequency is \(\omega / 2 \pi\text{.}\) These two functions share the additional property that their average value is zero. That is,

\begin{equation*}
a x'' + bx' + cx = A \cos \omega t + B \sin \omega t,
\end{equation*}

we can use Euler's formula, \(e^{i \beta t} = \cos \beta t + i \sin \beta t\) to derive a particular solution. That is, we will assume that our particular solution has the form

\begin{equation*}
x_c = x_\text{Re} + i x_\text{Im}
\end{equation*}

and use the properties of complex numbers.^{ 17 }If complex numbers make you uncomfortable, the alternative is to become an expert in trigonometric identities

Notice that all solutions of (4.14) will approach the particular solution as \(t \to \infty\text{.}\)

Example4.18

Now let us solve (4.14) using complex numbers. If we assume that the equation has a complex solution of the form \(x_c = x_\text{Re} + i x_\text{Im}\text{,}\) then

where \(\phi \approx 3.058451\text{.}\) We say that \(\phi\) is the phase angle of our solution. The amplitude of our solution is \(1/\sqrt{145}\) and the period is \(\pi\) (Figure 4.20).

The corresponding first order system for the differential equation

This is a nonautonomous system, and the tangent vector of a solution curve in the phase plane depends not only on the position \((x, y)\text{,}\) but also on the time \(t\text{.}\) In other words, the direction field changes with time. Since the direction field changes with time, two solutions with the same \((x,y)\) value at different times can follow different paths. Consequently, solutions can cross each other in the \(xy\)-plane without violating the Existence and Uniqueness Theorem.

Example4.21

Consider the harmonic oscillator that is modeled by the differential equation

and we will use the Method of Undetermined Coefficients and assume that we can find a particular solution of the form \(x_c = A e^{3it}\text{.}\) Substituting \(x_c\) into equation (4.15), we find that

\begin{equation*}
(8 + 6i) A e^{3it} = -2 e^{3it}.
\end{equation*}

Thus, \(x_c\) is a solution if

\begin{equation*}
A = \frac{-2}{8 + 6i} = - \frac{4}{25} + \frac{3}{25} i
\end{equation*}

The graph of our solution is given in Figure 4.22.

Since \(y = x'(t)\text{,}\) we can now graph the solution curve in the phase plane (Figure 4.23). Notice how the solution curve can intersect itself. The restoring force and damping are proportional to \(x\) and \(y = x'\text{,}\) respectively. When \(x\) and \(y\) are close to the origin, the external force is as large or larger than the restoring and damping forces. In this part of the \(xy\)-plane, the external force overcomes the damping and pushes the solution away from the origin.

On the other hand, suppose we have initial conditions \(x(0) = 2\) and \(x'(0) = 2\text{,}\) we can solve the linear system

The graph of our solution is given in Figure 4.24.

If we examine the phase plane for this solution (Figure 4.25), we see that the initial damping and restoring forces are much larger than the external force. Thus, if we are far from the origin, the solutions in the \(xy\)-plane tend to spiral towards the origin and are similar to the solutions of the unforced equation.

The functions \(\sin \omega t\) and \(g(t) = \cos \omega t\) are periodic with period \(2 \pi / \omega\) and frequency \(\omega / 2 \pi\text{.}\) These average value of each of these functions is zero.

We can use Euler's formula and complexification to solve the equation

where the forcing function \(g(t)\) is \(\sin \omega t\) or \(\cos \omega t\text{.}\) Furthermore, we can use complex numbers to express or solution in the form

\begin{equation*}
x(t) = A \cos(\omega t - \phi),
\end{equation*}

where \(A\) is the amplitude of the solution, \(\omega / 2 \pi\) is the frequency of the solution, and \(\phi\) is the phase angle.

we obtain a nonautonomous system. In this case the direction field changes with time, and two solutions with the same \((x,y)\) value at different times can follow different paths. Therefore, solutions can cross each other without violating the Existence and Uniqueness Theorem.

If we are far from the origin, the solutions in the \(xy\)-plane tend to spiral towards the origin and are similar to the solutions of the unforced equation. When \(x\) and \(y\) are close to the origin, the external force is as large or larger than the restoring and damping forces. In this part of the \(xy\)-plane, the external force overcomes the damping and pushes the solution away from the origin.