1.2: 1.2 Overview of the Course

For the most part, your first course in differential equations was about solving initial value problems. When second order equations did not fall into the above cases, then you might have learned how to obtain approximate solutions using power series methods, or even finding new functions from these methods. In this course we will explore two broad topics: systems of differential equations and boundary value problems.

We will see that there are interesting initial value problems when studying systems of differential equations. In fact, many of the second order equations that you have seen in the past can be written as a system of two first order equations. For example, the equation for simple harmonic motion,

\(x^{\prime \prime}+\omega^{2} x=0\),

can be written as the system

\begin{gathered}
x^{\prime}=y \\
y^{\prime}=-\omega^{2} x
\end{gathered}

Just note that \(x^{\prime \prime}=y^{\prime}=-\omega^{2} x\). Of course, one can generalize this to systems with more complicated right hand sides. The behavior of such systems can be fairly interesting and these systems result from a variety of physical models.
In the second part of the course we will explore boundary value problems. Often these problems evolve from the study of partial differential equations. Such examples stem from vibrating strings, temperature distributions, bending beams, etc. Boundary conditions are conditions that are imposed at more than one point, while for initial value problems the conditions are specified at one point. For example, we could take the oscillation equation above and ask when solutions of the equation would satisfy the conditions \(x(0)=0\) and \(x(1)=0\). The general solution, as we have determined earlier, is

\[x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t \nonumber \]

Requiring \(x(0)=0\), we find that \(c_{1}=0\), leaving \(x(t)=c_{2} \sin \omega t\). Also imposing that \(0=x(1)=c_{2} \sin \omega\), we are forced to make \(\omega=n \pi\), for \(n=1,2, \ldots\) (Making \(c_{2}=0\) would not give a nonzero solution of the problem.) Thus, there are an infinite number of solutions possible, if we have the freedom to choose our \(\omega\). In the second half of the course we will investigate techniques for solving boundary value problems and look at several applications, including seeing the connections with partial differential equations and Fourier series.