Differential Equations

A Modern Approach with Wavelets

Steven Krantz

• Publisher: CRC Press
• Year: 2020
• ISBN:
  9781000768466
• Edition: 1
• Topic: Mathématiques
• Subtopic: Équations différentielles

This new book from one of the most published authors in all of mathematics is an attempt to offer a new, more modern take on the Differential Equations course. The world is changing. Because of the theory of wavelets, Fourier analysis is ever more important and central. And applications are a driving force behind much of mathematics.This text text presents a more balanced picture. The text covers differential equations (both ordinary and partial), Fourier analysis and applications in equal measure and with equal weight. The Riemann integral is used throughout. We do not assume that the student knows any functional analysis. We likewise do not assume that the student has had a course in undergraduate real analysis. To make the book timely and exciting, a substantial chapter on basic properties of wavelets, with applications to signal processing and image processing is included. This should give students and instructors alike a taste of what is happening in the subject today.

1

What Is a Differential Equation?

• The concept of a differential equation

• Characteristics of a solution

• Finding a solution

• Separable equations

• First-order linear equations

• Exact equations

• Orthogonal trajectories

1.1 Introductory Remarks

A differential equation is an equation relating some function f to one or more of its derivatives. An example is

$\frac{d^{2}f}{d{x}^2}\left(x\right)+2x\frac{df}{dx}\left(x\right)+f^2\left(x\right)=\sin x.$

(1.1.1)

Observe that this particular equation involves a function f together with its first and second derivatives. Any given differential equation may or may not involve f or any particular derivative of f. But, for an equation to be a differential equation, at least some derivative of f must appear. The objective in solving an equation like (1.1.1) is to find the function f. Thus we already perceive a fundamental new paradigm: When we solve an algebraic equation, we seek a number or perhaps a collection of numbers; but when we solve a differential equation we seek one or more functions.

As a simple example, consider the differential equation

$y^{\prime }=y.$

It is easy to determine that any function of the form y = Ce^x is a solution of this equation. For the derivative of the function is equal to itself. So we see that the solution set of this particular differential equation is an infinite family of functions parametrized by a parameter C. This phenomenon is quite typical of what we will see when we solve differential equations in the chapters that follow.

Many of the laws of nature—in physics, in engineering, in chemistry, in biology, and in astronomy—find their most natural expression in the language of differential equations. Put in other words, differential equations are the language of nature. Applications of differential equations also a...