Synopses & Reviews
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.
- 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.
- 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