Differential Equations/Introduction

From Differential Equations

What are Differential Equations?

A differential equation is a relationship between an independent variable, (let us say x), a dependent variable (let us call this y), and one or more derivatives of y with respect to x. For example:

$xy\frac{d^{2}y}{d{x}^2}+y\frac{dy}{dx}+e^{3x}=0$

is a differential equation.

Why are they useful?

Remember that a derivative is the rate of change for some quantity with respect to another quantity. Frequently in science, you won't know an exact equation for some variable, but you may know its rate of change. Using differential equations you can work from that equation to the equation you really want. Differential equations reflect changing patterns in nature, history, and physics.

When can they occur?

Differential equations (DEs) often occur in systems when one variable is related to the rate of change of another, or vice versa. For example, the water level in a bucket with a hole in it being filled by a steady flow of water is governed by differential equations, as the rate of flow of water out of the bucket is proportional to the depth of the water.

In general, DEs may be formed from a consideration of the physical properties to which they refer. Often they occur when arbitrary constants are eliminated from a function. For example, suppose

$y(x)=a\sin x+b\cos x,$

with a and b being arbitrary constants. Differentiating this function twice (with respect to x) gives

$\frac{d^{2}y}{d{x}^2}=-a\sin x-b\cos x,$

which is equal to the negative variant of the original equation. Hence,

$\frac{d^{2}y}{d{x}^2}=-y.$

You may have noticed that the example involved the second derivative, which brings us to an important concept. The order of a differential equation is defined as the highest derivative order involved in the equation. This will be explained more fully later.

What should I already know to use this book?

Calculus is an absolute must. If you don't know how to do derivatives and integrals, you will not be able to follow this book. A basic understanding of trigonometry (mainly for identities to clean up answers and perform mathematical 'backflips' to make the problem more easily solvable) and complex numbers will also help. If you don't remember all your trig identities, don't worry — when I use a non-basic one for the first time, I will explain it. For the systems sections, a basic understanding of vectors and matrices is needed. The ability to multiply matrices and solve normal systems of equations via matrices will suffice.

Topics covered in this book

At current time, I plan on covering only ordinary differential equations, not partial differential equations. I may add partials later, if I feel I am capable of writing the text. Most of the time will be spent on linear differential equations. By the end of this text, you should be capable of solving the vast majority of solvable linear systems.

How to use this book

The book is divided into lessons, each lesson covering one topic. Each lesson has four parts:

• Part 1 of the lesson explains the concept and states any theorems, proofs, and instructions needed.
• Part 2 of the lesson is a set of real world uses for the technique — I hate not knowing why something is useful just as much as you do.
• Part 3 is a set of problems you can do for practice.
• Part 4 is the answers, with work, for part 3.

I strongly suggest starting the book at Lesson 1 and moving on. Each lesson builds on the last, and no time is given to the revision of previous topics. If you are reading this to refresh your knowledge of differential equations, be sure you really do remember the concepts, and be prepared to go back if you don't.

Conventions used in this book

Derivatives are specified in one of three ways, depending on what best suits this book's purpose:

• $\frac{dy}{dx}$
• $y^{\prime }$
• $y^{(3)}$ (mainly used for high order derivatives)

Important terms are put in bold the first time they are seen. All of these terms can be found in the glossary.