An Introduction to Differential Equations with Applications

The following sections are included:

• DEDICATION
• PREFACE
• CONTENTS

Ordinary differential equations involve derivatives of a function that depends on a single independent variable, which can be expressed as

That is, F is a function of the independent variable x, the dependent variable y and various derivatives of y with respect to x…

The general form of a first order differential equation is

Its solution contains one constant of integration, the value of which is determined from a given value of y. Such a constant is determined from a knowledge of the value of y at some value of x denoted by x₀. We refer to such a constraint as an initial condition, expressed in the form b ≡ y(a).

When an exact solution to a first order differential equation cannot be achieved using one of the methods developed in chapter 2, it is still possible in many cases, to obtain estimates of the solution using numerical approximation techniques. These methods yield numerical values of y(x) at selected values of x. In this chapter we present some of the most successful of these methods…

Many phenomena can be described by first order differential equations. We illustrate the application of first order differential equations by examples.

The following sections are included:

• Differential and Integral Operators
• Solutions to Linear Differential Equations with Constant Coefficients by Operator Methods
• The Homogeneous Solution
• Particular Solution When Q(x) is an Exponential
• Particular Solution When Q(x) is a Sum of Sine and Cosine
• Particular Solution When Q(x) is a Sum of Sinh and Cosh
• Particular Solution When Q(x) is a Polynomial
• Particular Solution When Q(x) is the Product of Exponential, Trigonometric and Hyperbolic Sines and Cosines and a Polynomial
• Evaluating Moment Integrals Using Operators
• Variation of Parameters
• Problems

The details involved in solving linear differential equations by series methods are applicable to equations of all orders. To introduce this development at the simplest level, this chapter is devoted to series methods for linear first order differential equations of the form

We note that the differential equation has been manipulated so that the coefficient of Dy is 1…

The general form of a linear second order differential equation is

With R(x) = 0, eq. 7.1 becomes the homogeneous second order differential equation

When the solution to a second order differential equation cannot be found in closed form, we can estimate the value of that solution y(x) at given values of x by approximation methods. Henceforth, for simplicity, we will denote Dy as y′, D²y as y″ and so on…

Many applications of the types of differential equation given in eq. 8.1 are in the field of physics, particularly in the areas of quantum mechanics and classical electromagnetic theory (including phenomena involving light, which is electromagnetic radiation). A list of such equations, is given below in table 9.1…

In this chapter, we define the Laplace Transform and derive several of the properties of these transforms. The reader who is familiar with the definition and properties of Laplace transforms can skip this chapter and move on to chapter 11 where we study the methods of solving differential equations by Laplace transform methods.

In chapter 10, several properties of Laplace transforms were derived (table 10.1) and several Laplace transform pairs were developed (table 10.2). With these, Laplace transform methods can be used to solve differential equations that are constrained by initial conditions…

The following sections are included:

• Coupled Algebraic Equations (A Review)
• Coupled First Order Differential Equations by Substitution
• Coupled First Order Differential Equations by Matrix Methods
• Coupled First Order Differential Equations by Laplace Transform Methods
• High Order Linear Differential Equations as Coupled First Order Linear Differential Equations
• Coupled Linear Second Order Differential Equations by Substitution
• Coupled Linear Second Order Differential Equations by Laplace Transform Methods
• Applications of Coupled Linear Second Order Differential Equations
• Problems

The following sections are included:

• By Taylor Series Methods
• By Finite Difference Methods
• By Interpolation Using the Volterra Method
• Problems

The following sections are included:

• Appendix 1. TAYLOR EXPANSION of a TWO-VARIABLE FUNCTION
• Appendix 2. CAUCHY RATIO for a MULTI-TERM RECURRENCE RELATION
• Appendix 3. AN RLC CIRCUIT
• LIST of EXAMPLES
• BIBLIOGRAPHY

This section contains answers to selected odd-numbered problems.

The following section is included:

• INDEX

The following sections are included:

• SUPPLEMENTAL MATERIALS CONTENTS
• CHAPTER 14: INTRODUCTION to SLOPE FIELD and PHASE PLANE ANALYSIS
• CHAPTER 15: INTRODUCTION to PARTIAL DIFFERENTIAL EQUATIONS
• CHAPTER 16: APPROXIMATION METHODS for PARTIAL DIFFERENTIAL EQUATIONS
• REVIEW SECTIONS