One way to specify a sequence of numbers is recursively; that is, the first one or more terms in the sequence are stated, and then a formula for the general term of the sequence is given in terms of one or more previous terms. For example, in the famous Fibonacci sequence, the first two terms … Read more
In the two previous posts in this series we explored a method for solving second order linear differential equations with constant coefficients that is different from the standard textbook methods taught nowadays. I found the method in a 1941 book (or see here) by the Sokolnikoffs. The key point of the method, as we learned, … Read more
In a previous post we discussed an operator method for solving certain second order ordinary differential equations. In this post I’ll explore this operator method a little further. I first learned about this method from an old book, Higher Mathematics for Engineers and Physicists, by Ivan S. Sokolnikoff and Elizabeth S. Sokolnikoff, McGraw-Hill, 1941. I … Read more
In talking about power series in a previous post, I mentioned one of their uses: as an aid in solving differential equations. This reminds me of a neat trick for solving some differential equations, which I will discuss in this post. A standard method for solving linear differential equations with constant coefficients is to assume … Read more
We’ve been talking about reductionism in the past couple of posts, and we’ll continue the story by discussing power series in this post. The idea behind reductionism in mathematics is to identify some elementary “objects” and to express a complicated “thing” in terms of the elementary things. The intention is either to make it easier … Read more