Ordinary Differential Equations/Introduction

What are Differential Equations?

The term differential equation was coined by Leibniz in 1676 for a relationship between the two differentials dx and dy for the two variables x and y.[citation needed]

However, soon after the first usage of this term, differential equations quickly became understood as any algebraic or transcendental equation which involved derivatives.

An ordinary differential equation is a relationship between a real variable, (let us say x), a dependent variable (let us call this y), and (possibly many) derivatives of the dependent variable y with respect to x. For example:

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

is an ordinary differential equation.

A partial differential equation is a relationship between a function of several variables, the partial derivatives of that function, and the independent variables. Usually they are much more difficult to deal with than ordinary differential equations.

$\frac{\partial^2 u}{\partial x^2}(x,y)+x^2\frac{\partial^2 u}{\partial y^2}(x,y)=0$

is an example of a partial differential equation.

Objectives

When given a ordinary differential equations problem, the objective is typically to find all functions y(x), so that this function and its derivatives satisfy the relationship specified by the equation. In other words if you plug the function into the equation you arrive at an equality. Explicitly finding such functions in terms of functions we already know is frequently not possible. When it is possible it may involve creative mathematical techniques. However many standard techniques exist for some of the more common types of ordinary differential equation and they will be explained in this Wikibook. In cases where explicit solutions do not exist, one is left to investigate properties of the function that may be deduced entirely from the equation. Examples of such questions might be

• "Do solutions exist at all?"
• "Is there just one solution, or many, and can they be classified?"
• "How can we approximate a solution numerically?"
• "What is the asymptotic behavior of a solution"? (That is how does it behave as, say x → ∞).

The subject of differential equations is built upon the subject of calculus. The reader is expected to be familiar with Multivariable calculus, though for the vast majority of topics discussed knowledge of single variable calculus will suffice. Particularly when systems of ordinary differential equations are discussed it will also be necessary to assume the reader is familiar with linear algebra. Though for much of the book this will also not be necessary to read the sections.

Why are they useful?

Differential equations occur frequently in many branches of science and in both pure and applied mathematics. One possible explanation for this is to remember that a derivative describes a rate of change, so anytime it is useful to describe how changes in one thing depend on changes in some other thing, differential equations are lurking in the background (or possibly foreground!). Differential equations allow us to model changing patterns in both physical and mathematical problems.

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 can be described 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

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

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

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

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:

• $\frac{dy}{dx}$
• $y' \,$
• $y^{(3)} \,$ (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.