﻿ why study differential equations with linear algebra?

# why study differential equations with linear algebra?

In elementary school we memorize algorithms for multiplying and dividing multidigit numbers as part of our acquiring a coherent picture of mathematical thinking at that level, but in practice later we always use calculators or software to do these mechanical tasks. Of course having a sense of what approximate results to expect is important to avoid believing obviously incorrect results which result from improper input ("garbage in, garbage out"). "Rule of thumb" or "ballpark number reasoning" estimates serve this purpose.

When we study calculus of a single variable, the end goal is not to learn rules of differentiation and integration but to understand how those mechanical operations can be used with specific functions to understand their behavior. Yes, it is important to learn these rules as one ingests and digests the concepts of calculus in order to acquire a complete understanding of what you are doing, but once you understand the ideas and how to use them in analyzing functional relationships, the mechanical calculations can be done more effectively by a computer algebra system, which also can handle difficult calculations that cannot be done by hand. Solving equations along the way are also mechanical calculations which software can do for us, especially since in more realistic problems (Nature is complicated!), we cannot use our algebra skills to solve those equations but must rely on software to do so either symbolically or numerically.

This course continues the calculus of a single independent variable to its application in understanding ordinary differential equations (ODEs), while multivariable calculus with multiple independent variables follows up with partial differential equations (PDEs). You will most probably encounter the latter topic not in any actual PDE math course, but in applications in applied math in STEM fields. The quantum mechanical Schroedinger's equation underlies all of chemistry, for example. Heat flow in engineering. Maxwell's equations in physics, which govern all the electromagnetic devices and communication services our modern civilization relies on. Ordinary and partial differential equations govern most of the laws of Nature.

Ordinary differential equations, or just DEs from now on since we are only concerned with these, are simply equations involving one or more unknown functions of a single independent variable and a finite number of their derivatives. In practice mostly first and second derivatives are enough to describe Nature. Nature has been kind to us because most of the differential equations we need are actually "linear" in the unknowns and their derivatives, and thus allow vector and matrix operations to more efficiently calculate and interpret solutions. Thus we will study enough linear mathematics to gain this efficiency not only in calculating, but in visualizing what we are calculating and interpreting it. Linearity allows us to add simple solutions together to make more complicated ones, like superimposing wave fronts emanating outwards from multiple stones tossed into a calm pond---the waves pass right through each other.

The actual methods of "solving" DEs are not our end goal, exactly analogous to why rules of differentiation and integration are not the end goal of one variable calculus. Software can do this much better than we can, but for us to understand how it does this and why it produces the results it does, we need to learn a few basic techniques for solving DEs. This will then give us familiarity with the kinds of functional relationships which come out of these solution processes and which describe basic features of Nature. Oscillations (trigonometric functions) and growth and decay (exponential functions) are absolutely fundamental to this process of developing an understanding of how we can use what we learn in this course in then understanding applications in STEM field courses that attempt to explain Nature.

Every really tall building (skyscrapers) sways back and forth under wind forces. This is only one of many vibrational modes, the lowest frequency in fact. Like a plucked guitar string, it has many modes, and when an earthquake shakes a building laterally, these other modes can be excited to "break it", push it beyond its flexible vibrational limits. The techniques we will study in matrix mathematics will help us be able to calculate this behavior at the end of the course, even if we do not "cover" this particular example in our testable material. By understanding just a system with two independent variables (lateral positions of the two ceilings in a two story building), we can make the leap to any number of variables, i.e., a multistory building of any number of stories. In that way a simple math model is the gateway to a more complicated analysis specialists (engineers!) can carry out. Similarly a simple coupling of two "damped" oscillating systems shows the physical principles underlying designing stability into skyscrapers that involve hundreds of floors.

Here is an example taken from our textbook about earthquake modeling of tall buildings. The slowest "vibration" is the swaying back and forth motion.

And a 9 minute video of a simple oscillator experiment shows how tall buildings can be engineered to damp out those swaying oscillations.