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Differential Equations Problem Solver

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Differential Equations Problem Solver Cover

ISBN13: 9780878915132
ISBN10: 0878915133
Condition: Standard
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Synopses & Reviews

Publisher Comments:

Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.

Here in this highly useful reference is the finest overview of differential equations currently available, with hundreds of differential equations problems that cover everything from integrating factors and Bernoulli's equation to variation of parameters and undetermined coefficients. Each problem is clearly solved with step-by-step detailed solutions.

DETAILS

- The PROBLEM SOLVERS are unique - the ultimate in study guides.

- They are ideal for helping students cope with the toughest subjects.

- They greatly simplify study and learning tasks.

- They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding.

- They cover material ranging from the elementary to the advanced in each subject.

- They work exceptionally well with any text in its field.

- PROBLEM SOLVERS are available in 41 subjects.

- Each PROBLEM SOLVER is prepared by supremely knowledgeable experts.

- Most are over 1000 pages.

- PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly.

TABLE OF CONTENTS

Introduction

Units Conversion Factors

Chapter 1: Classification of Differential Equations

Chapter 2: Separable Differential Equations

Variable Transformation u = ax + by

Variable Transformation y = vx

Chapter 3: Exact Differential Equations

Definitions and Examples

Solving Exact Differential Equations

Making a Non-exact Differential Equation Exact

Chapter 4: Homogenous Differential Equations

Identifying Homogenous Differential Equations

Solving Homogenous Differential Equations by Substitution and Separation

Chapter 5: Integrating Factors

General Theory of Integrating Factors

Equations of Form dy/dx + p(x)y = q(x)

Grouping to Simplify Solutions

Solution Directly From M(x,y)dx + N(x,y)dy = 0

Chapter 6: Method of Grouping

Chapter 7: Linear Differential Equations

Integrating Factors

Bernoulli's Equation

Chapter 8: Riccati's Equation

Chapter 9: Clairaut's Equation

Geometrical Construction Problems

Chapter 10: Orthogonal Trajectories

Elimination of Constants

Orthogonal Trajectories

Differential Equations Derived from Considerations of Analytical Geometry

Chapter 11: First Order Differential Equations: Applications I

Gravity and Projectile

Hooke's Law, Springs

Angular Motion

Over-hanging Chain

Chapter 12: First Order Differential Equations: Applications II

Absorption of Radiation

Population Dynamics

Radioactive Decay

Temperature

Flow from an Orifice

Mixing Solutions

Chemical Reactions

Economics

One-Dimensional Neutron Transport

Suspended Cable

Chapter 13: The Wronskian and Linear Independence

Determining Linear Independence of a Set of Functions

Using the Wronskian in Solving Differential Equations

Chapter 14: Second Order Homogenous Differential Equations with Constant Coefficients

Roots of Auxiliary Equations: Real

Roots of Auxiliary: Complex

Initial Value

Higher Order Differential Equations

Chapter 15: Method of Undetermined Coefficients

First Order Differential Equations

Second Order Differential Equations

Higher Order Differential Equations

Chapter 16: Variation of Parameters

Solution of Second Order Constant Coefficient Differential Equations

Solution of Higher Order Constant Coefficient Differential Equations

Solution of Variable Coefficient Differential Equations

Chapter 17: Reduction of Order

Chapter 18: Differential Operators

Algebra of Differential Operators

Properties of Differential Operators

Simple Solutions

Solutions Using Exponential Shift

Solutions by Inverse Method

Solution of a System of Differential Equations

Chapter 19: Change of Variables

Equation of Type (ax + by + c)dx + (dx + ey + f)dy = 0

Substitutions for Euler Type Differential Equations

Trigonometric Substitutions

Other Useful Substitutions

Chapter 20: Adjoint of a Differential Equation

Chapter 21: Applications of Second Order Differential Equations

Harmonic Oscillator

Simple Pendulum

Coupled Oscillator and Pendulum

Motion

Beam and Cantilever

Hanging Cable

Rotational Motion

Chemistry

Population Dynamics

Curve of Pursuit

Chapter 22: Electrical Circuits

Simple Circuits

RL Circuits

RC Circuits

LC Circuits

Complex Networks

Chapter 23: Power Series

Some Simple Power Series

Solutions May Be Expanded

Finding Power Series Solutions

Power Series Solutions for Initial Value Problems

Chapter 24: Power Series about an Ordinary Point

Initial Value Problems

Special Equations

Taylor Series Solution to Initial Value Problem

Chapter 25: Power Series about a Singular Point

Singular Points and Indicial Equations

Frobenius Method

Modified Frobenius Method

Indicial Roots: Equal

Special Equations

Chapter 26: Laplace Transforms

Exponential Order

Simple Functions

Combination of Simple Functions

Definite Integral

Step Functions

Periodic Functions

Chapter 27: Inverse Laplace Transforms

Partial Fractions

Completing the Square

Infinite Series

Convolution

Chapter 28: Solving Initial Value Problems by Laplace Transforms

Solutions of First Order Initial Value Problems

Solutions of Second Order Initial Value Problems

Solutions of Initial Value Problems Involving Step Functions

Solutions of Third Order Initial Value Problems

Solutions of Systems of Simultaneous Equations

Chapter 29: Second Order Boundary Value Problems

Eigenfunctions and Eigenvalues of Boundary Value Problem

Chapter 30: Sturm-Liouville Problems

Definitions

Some Simple Solutions

Properties of Sturm-Liouville Equations

Orthonormal Sets of Functions

Properties of the Eigenvalues

Properties of the Eigenfunctions

Eigenfunction Expansion of Functions

Chapter 31: Fourier Series

Properties of the Fourier Series

Fourier Series Expansions

Sine and Cosine Expansions

Chapter 32: Bessel and Gamma Functions

Properties of the Gamma Function

Solutions to Bessel's Equation

Chapter 33: Systems of Ordinary Differential Equations

Converting Systems of Ordinary Differential Equations

Solutions of Ordinary Differential Equation Systems

Matrix Mathematics

Finding Eigenvalues of a Matrix

Converting Systems of Ordinary Differential Equations into Matrix Form

Calculating the Exponential of a Matrix

Solving Systems by Matrix Methods

Chapter 34: Simultaneous Linear Differential Equations

Definitions

Solutions of 2 x 2 Systems

Checking Solution and Linear Independence in Matrix Form

Solution of 3 x 3 Homogenous System

Solution of Non-homogenous System

Chapter 35: Method of Perturbation

Chapter 36: Non-Linear Differential Equations

Reduction of Order

Dependent Variable Missing

Independent Variable Missing

Dependent and Independent Variable Missing

Factorization

Critical Points

Linear Systems

Non-Linear Systems

Liapunov Function Analysis

Second Order Equation

Perturbation Series

Chapter 37: Approximation Techniques

Graphical Methods

Successive Approximation

Euler's Method

Modified Euler's Method

Chapter 38: Partial Differential Equations

Solutions of General Partial Differential Equations

Heat Equation

Laplace's Equation

One-Dimensional Wave Equation

Chapter 39: Calculus of Variations

Index

WHAT THIS BOOK IS FOR

Students have generally found differential equations a difficult subject to understand and learn. Despite the publication of hundreds of textbooks in this field, each one intended to provide an improvement over previous textbooks, students of differential equations continue to remain perplexed as a result of numerous subject areas that must be remembered and correlated when solving problems. Various interpretations of differential equations terms also contribute to the difficulties of mastering the subject.

In a study of differential equations, REA found the following basic reasons underlying the inherent difficulties of differential equations:

No systematic rules of analysis were ever developed to follow in a step-by-step manner to solve typically encountered problems. This results from numerous different conditions and principles involved in a problem that leads to many possible different solution methods. To prescribe a set of rules for each of the possible variations would involve an enormous number of additional steps, making this task more burdensome than solving the problem directly due to the expectation of much trial and error.

Current textbooks normally explain a given principle in a few pages written by a differential equations professional who has insight into the subject matter not shared by others. These explanations are often written in an abstract manner that causes confusion as to the principle's use and application. Explanations then are often not sufficiently detailed or extensive enough to make the reader aware of the wide range of applications and different aspects of the principle being studied. The numerous possible variations of principles and their applications are usually not discussed, and it is left to the reader to discover this while doing exercises. Accordingly, the average student is expected to rediscover that which has long been established and practiced, but not always published or adequately explained.

The examples typically following the explanation of a topic are too few in number and too simple to enable the student to obtain a thorough grasp of the involved principles. The explanations do not provide sufficient basis to solve problems that may be assigned for homework or given on examinations.

Poorly solved examples such as these can be presented in abbreviated form which leaves out much explanatory material between steps, and as a result requires the reader to figure out the missing information. This leaves the reader with an impression that the problems and even the subject are hard to learn - completely the opposite of what an example is supposed to do.

Poor examples are often worded in a confusing or obscure way. They might not state the nature of the problem or they present a solution, which appears to have no direct relation to the problem. These problems usually offer an overly general discussion - never revealing how or what is to be solved.

Many examples do not include accompanying diagrams or graphs, denying the reader the exposure necessary for drawing good diagrams and graphs. Such practice only strengthens understanding by simplifying and organizing differential equations processes.

Students can learn the subject only by doing the exercises themselves and reviewing them in class, obtaining experience in applying the principles with their different ramifications.

In doing the exercises by themselves, students find that they are required to devote considerable more time to differential equations than to other subjects, because they are uncertain with regard to the selection and application of the theorems and principles involved. It is also often necessary for students to discover those "tricks" not revealed in their texts (or review books) that make it possible to solve problems easily. Students must usually resort to methods of trial and error to discover these "tricks," therefore finding out that they may sometimes spend several hours to solve a single problem.

When reviewing the exercises in classrooms, instructors usually request students to take turns in writing solutions on the boards and explaining them to the class. Students often find it difficult to explain in a manner that holds the interest of the class, and enables the remaining students to follow the material written on the boards. The remaining students in the class are thus too occupied with copying the material off the boards to follow the professor's explanations.

This book is intended to aid students in differential equations overcome the difficulties described by supplying detailed illustrations of the solution methods that are usually not apparent to students. Solution methods are illustrated by problems that have been selected from those most often assigned for class work and given on examinations. The problems are arranged in order of complexity to enable students to learn and understand a particular topic by reviewing the problems in sequence. The problems are illustrated with detailed, step-by-step explanations, to save the students large amounts of time that is often needed to fill in the gaps that are usually found between steps of illustrations in textbooks or review/outline books.

The staff of REA considers differential equations a subject that is best learned by allowing students to view the methods of analysis and solution techniques. This learning approach is similar to that practiced in various scientific laboratories, particularly in the medical fields.

In using this book, students may review and study the illustrated problems at their own pace; students are not limited to the time such problems receive in the classroom.

When students want to look up a particular type of problem and solution, they can readily locate it in the book by referring to the index that has been extensively prepared. It is also possible to locate a particular type of problem by glancing at just the material within the boxed portions. Each problem is numbered and surrounded by a heavy black border for speedy identification.

Synopsis:

Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.

Here in this highly useful reference is the finest overview of differential equations currently available, with hundreds of differential equations problems that cover everything from integrating factors and Bernoulli's equation to variation of parameters and undetermined coefficients. Each problem is clearly solved with step-by-step detailed solutions.

DETAILS

- The PROBLEM SOLVERS are unique - the ultimate in study guides.

- They are ideal for helping students cope with the toughest subjects.

- They greatly simplify study and learning tasks.

- They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding.

- They cover material ranging from the elementary to the advanced in each subject.

- They work exceptionally well with any text in its field.

- Each PROBLEM SOLVER is prepared by supremely knowledgeable experts.

- Most are over 1000 pages.

- PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly.

Table of Contents

TABLE OF CONTENTS

Introduction

Units Conversion Factors

Chapter 1: Classification of Differential Equations

Chapter 2: Separable Differential Equations

Variable Transformation u = ax + by

Variable Transformation y = vx

Chapter 3: Exact Differential Equations

Definitions and Examples

Solving Exact Differential Equations

Making a Non-exact Differential Equation Exact

Chapter 4: Homogenous Differential Equations

Identifying Homogenous Differential Equations

Solving Homogenous Differential Equations by Substitution and Separation

Chapter 5: Integrating Factors

General Theory of Integrating Factors

Equations of Form dy/dx + p(x)y = q(x)

Grouping to Simplify Solutions

Solution Directly From M(x,y)dx + N(x,y)dy = 0

Chapter 6: Method of Grouping

Chapter 7: Linear Differential Equations

Integrating Factors

Bernoulli's Equation

Chapter 8: Riccati's Equation

Chapter 9: Clairaut's Equation

Geometrical Construction Problems

Chapter 10: Orthogonal Trajectories

Elimination of Constants

Orthogonal Trajectories

Differential Equations Derived from Considerations of Analytical Geometry

Chapter 11: First Order Differential Equations: Applications I

Gravity and Projectile

Hooke's Law, Springs

Angular Motion

Over-hanging Chain

Chapter 12: First Order Differential Equations: Applications II

Absorption of Radiation

Population Dynamics

Radioactive Decay

Temperature

Flow from an Orifice

Mixing Solutions

Chemical Reactions

Economics

One-Dimensional Neutron Transport

Suspended Cable

Chapter 13: The Wronskian and Linear Independence

Determining Linear Independence of a Set of Functions

Using the Wronskian in Solving Differential Equations

Chapter 14: Second Order Homogenous Differential Equations with Constant Coefficients

Roots of Auxiliary Equations: Real

Roots of Auxiliary: Complex

Initial Value

Higher Order Differential Equations

Chapter 15: Method of Undetermined Coefficients

First Order Differential Equations

Second Order Differential Equations

Higher Order Differential Equations

Chapter 16: Variation of Parameters

Solution of Second Order Constant Coefficient Differential Equations

Solution of Higher Order Constant Coefficient Differential Equations

Solution of Variable Coefficient Differential Equations

Chapter 17: Reduction of Order

Chapter 18: Differential Operators

Algebra of Differential Operators

Properties of Differential Operators

Simple Solutions

Solutions Using Exponential Shift

Solutions by Inverse Method

Solution of a System of Differential Equations

Chapter 19: Change of Variables

Equation of Type (ax + by + c)dx + (dx + ey + f)dy = 0

Substitutions for Euler Type Differential Equations

Trigonometric Substitutions

Other Useful Substitutions

Chapter 20: Adjoint of a Differential Equation

Chapter 21: Applications of Second Order Differential Equations

Harmonic Oscillator

Simple Pendulum

Coupled Oscillator and Pendulum

Motion

Beam and Cantilever

Hanging Cable

Rotational Motion

Chemistry

Population Dynamics

Curve of Pursuit

Chapter 22: Electrical Circuits

Simple Circuits

RL Circuits

RC Circuits

LC Circuits

Complex Networks

Chapter 23: Power Series

Some Simple Power Series

Solutions May Be Expanded

Finding Power Series Solutions

Power Series Solutions for Initial Value Problems

Chapter 24: Power Series about an Ordinary Point

Initial Value Problems

Special Equations

Taylor Series Solution to Initial Value Problem

Chapter 25: Power Series about a Singular Point

Singular Points and Indicial Equations

Frobenius Method

Modified Frobenius Method

Indicial Roots: Equal

Special Equations

Chapter 26: Laplace Transforms

Exponential Order

Simple Functions

Combination of Simple Functions

Definite Integral

Step Functions

Periodic Functions

Chapter 27: Inverse Laplace Transforms

Partial Fractions

Completing the Square

Infinite Series

Convolution

Chapter 28: Solving Initial Value Problems by Laplace Transforms

Solutions of First Order Initial Value Problems

Solutions of Second Order Initial Value Problems

Solutions of Initial Value Problems Involving Step Functions

Solutions of Third Order Initial Value Problems

Solutions of Systems of Simultaneous Equations

Chapter 29: Second Order Boundary Value Problems

Eigenfunctions and Eigenvalues of Boundary Value Problem

Chapter 30: Sturm-Liouville Problems

Definitions

Some Simple Solutions

Properties of Sturm-Liouville Equations

Orthonormal Sets of Functions

Properties of the Eigenvalues

Properties of the Eigenfunctions

Eigenfunction Expansion of Functions

Chapter 31: Fourier Series

Properties of the Fourier Series

Fourier Series Expansions

Sine and Cosine Expansions

Chapter 32: Bessel and Gamma Functions

Properties of the Gamma Function

Solutions to Bessel's Equation

Chapter 33: Systems of Ordinary Differential Equations

Converting Systems of Ordinary Differential Equations

Solutions of Ordinary Differential Equation Systems

Matrix Mathematics

Finding Eigenvalues of a Matrix

Converting Systems of Ordinary Differential Equations into Matrix Form

Calculating the Exponential of a Matrix

Solving Systems by Matrix Methods

Chapter 34: Simultaneous Linear Differential Equations

Definitions

Solutions of 2 x 2 Systems

Checking Solution and Linear Independence in Matrix Form

Solution of 3 x 3 Homogenous System

Solution of Non-homogenous System

Chapter 35: Method of Perturbation

Chapter 36: Non-Linear Differential Equations

Reduction of Order

Dependent Variable Missing

Independent Variable Missing

Dependent and Independent Variable Missing

Factorization

Critical Points

Linear Systems

Non-Linear Systems

Liapunov Function Analysis

Second Order Equation

Perturbation Series

Chapter 37: Approximation Techniques

Graphical Methods

Successive Approximation

Euler's Method

Modified Euler's Method

Chapter 38: Partial Differential Equations

Solutions of General Partial Differential Equations

Heat Equation

Laplace's Equation

One-Dimensional Wave Equation

Chapter 39: Calculus of Variations

Index

What Our Readers Are Saying

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pramodpkanni020, August 30, 2011 (view all comments by pramodpkanni020)
Problem solvers by REA publications are well managed sum up for most of the Engineering topics. I come across after solving Differential equations from many books and found differential equation problem solver from google books as a sum up of all.
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salma-nasir, December 7, 2009 (view all comments by salma-nasir)
tooooo gud buk
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Product Details

ISBN:
9780878915132
Author:
Rea
Publisher:
Research & Education Association
Other:
Ogden, James
Author:
The Staff of Rea Good
Author:
The Staff of Rea
Author:
Arterburn, David R.
Author:
Harris, Kymberly
Author:
Ogden, James R.
Author:
Technology
Author:
REA, The Editors of
Author:
Research &. Education Association
Author:
The Staff of Rea
Author:
REA, Editors of
Location:
Piscataway, N.J. :
Subject:
Mathematics
Subject:
Differential Equations
Subject:
Problems, exercises, etc.
Subject:
Differential equations -- Problems, exercises, etc.
Subject:
General Mathematics
Subject:
Mathematics-Differential Equations
Copyright:
Edition Number:
Rev.
Edition Description:
Trade Paper
Series:
Problem Solvers Solution Guides
Publication Date:
January 2000
Binding:
TRADE PAPER
Language:
English
Illustrations:
Y
Pages:
1560
Dimensions:
10 x 6.69 in 4.5 lb
Age Level:
from 16

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Differential Equations Problem Solver Used Trade Paper
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Product details 1560 pages Research & Education Association - English 9780878915132 Reviews:
"Synopsis" by ,
Each Problem Solver is an insightful and essential study and solution guide chock-full of clear, concise problem-solving gems. All your questions can be found in one convenient source from one of the most trusted names in reference solution guides. More useful, more practical, and more informative, these study aids are the best review books and textbook companions available. Nothing remotely as comprehensive or as helpful exists in their subject anywhere. Perfect for undergraduate and graduate studies.

Here in this highly useful reference is the finest overview of differential equations currently available, with hundreds of differential equations problems that cover everything from integrating factors and Bernoulli's equation to variation of parameters and undetermined coefficients. Each problem is clearly solved with step-by-step detailed solutions.

DETAILS

- The PROBLEM SOLVERS are unique - the ultimate in study guides.

- They are ideal for helping students cope with the toughest subjects.

- They greatly simplify study and learning tasks.

- They enable students to come to grips with difficult problems by showing them the way, step-by-step, toward solving problems. As a result, they save hours of frustration and time spent on groping for answers and understanding.

- They cover material ranging from the elementary to the advanced in each subject.

- They work exceptionally well with any text in its field.

- Each PROBLEM SOLVER is prepared by supremely knowledgeable experts.

- Most are over 1000 pages.

- PROBLEM SOLVERS are not meant to be read cover to cover. They offer whatever may be needed at a given time. An excellent index helps to locate specific problems rapidly.

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