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,
We say that sinusoidal forcing occurs in the differential equation
\begin{equation*}
x'' + px' + qx = A \cos \omega t + B \sin \omega t.
\end{equation*}
Subsection2.3.1Complexification
Given a second-order linear differential equation
\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.
In the next examples, we solve a differential equation with sinusoidal forcing in two ways: first, using the method of undetermined coefficients, and second using complexification.
By our prior observations, \(x_{\text{Im}}\) is a solution to \(x'' + 6x' + 5x = \sin 2t\text{.}\) (Note that this agrees with what we found in Example 2.3.1.)
Activity2.3.1.Second-Order Linear Differential Equations and Complexification.
Find (1) a particular solution and (2) a general solution for each of the following differential equations.
(a)
\(x'' + 4x' - 21x = 2 e^{4t}\)
(b)
\(x'' + 4x' - 21x = 3e^{3t}\)
(c)
\(x'' - 4x' + 20x = 3 \sin 3t\)
(d)
\(x'' - 4x' + 20x = 2 e^{2t} \cos 4t\)
(e)
\(x'' - 14 x' + 49 x = \sin 3t\)
Subsection2.3.2Qualitative Analysis
We can use the complex solution of \(a x'' + bx' + cx = A \cos \omega t + B \sin \omega t\) to analyze the qualitative behavior of solutions. In the next example, we demonstrate how to write particular solutions to this equation in the compact form \(x_{p} = A\cos(\omega t - \phi)\text{,}\) where \(A\) is the amplitude of the solution, \(\omega\) is the frequency, and \(\phi\) is the phase angle.
Example2.3.3.
We discovered in Example 2.3.2 that the complex solution of
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 2.3.4).
an oscillating steady state curve plotted against time
Figure2.3.4.Steady state solution to \(x'' + 6 x' + 5x = \sin 2t\)
Activity2.3.2.Finding Particular Solutions of the Form \(y_p = A \cos(\omega t - \phi)\).
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 (2.3.4), 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*}
Since \(y = x'(t)\text{,}\) we can now graph the solution curve in the phase plane (Figure 2.3.7). 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.
the solution curve plotted in the phase plane where velocity is plotted against posiiotn and the solution curve spirals out towards a circular trajectory
If we examine the phase plane for this solution (Figure 2.3.9), 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 solution curve plotted in the phase plane where velocity is plotted against position and the solution curve spirals in towards a circular trajectory
The functions \(\sin \omega t\) and \(\cos \omega t\) 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 our 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.
Reading Questions2.3.4Reading Questions
1.
What does complexification mean?
2.
Is it possible for solution curves to intersect in the phase plane of a nonautonomous system? Why or why not?
Finding Frequencies, Amplitudes, and Phase Angles.
Find a particular solution of the form \(y_p = A \cos(\omega t - \phi)\) for each equation in Exercise Group 2.3.5.11–17 and determine the frequency \(\omega\text{,}\) amplitude \(A\text{,}\) and phase angle \(\phi\) of the solution.
11.
\(y'' + 4y = 3 \cos 2t\)
12.
\(y'' + 7y' + 10 y = - 4 \sin 3t\)
Hint.
Assume the complex solution has form \(y_c = A e^{3it}\text{.}\)
