Differential Equations and Linear Algebra: Why this Course?

Differential Equations and Linear Algebra are each separately important in applications of mathematics to physical problems, but they are especially important in linear differential equations where both topics play a fundamental role.

Most of the differential equations which describe physical phenomena are linear, allowing a superposition of solutions, like the constructive and destructive interference of wave patterns on water or in the air or of electromagnetic communications signals. Linear algebra is the branch of  mathematics which deals with the "linearity properties" of such problems, first encountered with vectors and matrices and with the sum and constant factor rules of differentiation in calculus.

Differential equations are equations which determine relationships between physical variables through conditions on their rates of change, i.e., derivatives with respect to other variables. The "ordinary" differential equations studied in this course involve derivative conditions on unknown functions of a single independent variable, often the time in physical problems, but open to any interpretation depending on the application.

Most of the equations governing the behavior of continuous physical variables whose values change in time are differential equations, and so describe all kinds of problems in nature, many of which we even have some intuition about. This course will show how to solve simple enough differential equations, and use them to allow us to gain some familiarity with general properties of differential equations and their solutions which are useful even for more complicated differential equations that we cannot find simple solutions for.

Warning

Like integration methods which are for the most part a waste of time these days, techniques for solving particular differential equations are not so important as the light they can shed about the way the solutions of the type of differential equation behave and the kinds of functions to expect as solutions for certain differential equations. Technology can solve any differential equation that we can solve by hand (sometimes it requires a little user help). We need to understand instead a bit about the types of functions that often pop up in solutions of simple differential equations, since in applications, the corresponding behavior provides us with the example physical systems that help develop our intuition about how nature works.