I am excited to teach our Sophomore-level ODE (Ordinary Differential Equations) course. When I came to City Tech 11 years ago, I taught this course quite often, almost every semester. After a few years, I began to teach the course less frequently. This is the first semester I am teaching the course using our current textbook. Given how fast textbooks become out of date and out of favor, we are surprisingly using the textbook I used when I took the course 25 years ago, albeit a much later edition (NOTE: I had little input into which textbook was chosen). In fact, much has changed since I took the course. Technology is now ubiquitous — direction fields and sample solutions can readily be displayed for order 1 ODE’s — so that we can and will turn our attention to qualitative issues much more frequently than when I took the course.
Solving ODE’s is perhaps the most direct and important application of the integration techniques you learned in Calculus II. I hope that you begin to reap some of the fruits resulting from all the hard work that you did to get through that course.
While being able to crank out solutions is important, it is not enough to do well in this course; you must be able to analyze and describe the long term behavior of solutions. Doing so, you will make use of deep mathematics begun in the 17th century and continuing into the 20th century. While ODE’s are now well understood, PDE’s (partial differential equations) make’s extensive use of ODE’s and is an active research area. It is perhaps the most important area of applied mathematics.