In this unit, we finish our study of differential equations. This includes a small section on second-order equations. The Briggs textbook includes an electronic-only chapter on these and we include it here. We will only be looking at the first two sections of that chapter (and, even then, only the aspects covered in the lecture videos).
We also begin to look at sequences and series. This part of the course starts with some very simple ideas and quickly builds into some very intricate concepts as we move into Taylor and other power series.
- Section 8.5: Obviously, we can do no more than scratch the surface of this concept. You will be finding out how to apply differential equations in your particular fields of engineering over the next few years, so we only introduce some of the main variants here. The questions about mixing solutions, while they most clearly relate to chemical engineering, are excellent for an understanding of what you can expect in the future.
- Extra: This section peeks at second-order DEs. It gives us a sense of what to expect in future years and a better idea of the way that linear algebra operates to create the solutions for differential equations. This section can be taken as mere memorization, or it can be studied in order to have a better understanding of the subject. I strongly recommend taking an hour or two to try to get a sense of how important linear algebra is within the context of calculus.
- Section 9.1: With this section, we start a long road to understanding how useful series are in calculus. Most of you will extend the ideas we end with - Taylor series - into the complex numbers next year, while some of you (especially those in ECE) will go even further with Fourier series.