Ordinary and Partial Differential Equations Texts

[Mathgician]

I was wondering what you guys think of my textbook.
My textbook is called:

A First Course in Differential Eqations (Eight Edition)

Author: Dennis G. Zill

ISBN: 0534418783

I have been using this book for my DE class, and I do not feel like I am really learning anything. This class is very different from my other math classes, I learn methods that I am not getting a solid explanations for. Most people I've gone for help that has taken this class before have forgotten most of the materials that has been tought in this class. I want to get the most out of my DE class, is there another book you guys suggest that will help me understand DE and not forget it?

 

[Astronuc]

I came across there and was wondering if anyone has had any experience with them.

https://www.amazon.com/dp/0127843965/?tag=pfamazon01-20
by Daniel Zwillinger
Academic Press; 3 edition (January 15, 1998)

This book and CD-ROM compile the most widely applicable methods for solving and approximating differential equations. The CD-ROM provides convenient access to these methods through electronic search capabilities, andtogether the book and CD-ROM contain numerous examples showing the methods use. Topics include ordinary differential equations, symplectic integration of differential equations, and the use of wavelets when numerically solving differential equations.

* For nearly every technique, the book and CD-ROM provide:
* The types of equations to which the method is applicable
* The idea behind the method
* The procedure for carrying out the method
* At least one simple example of the method
* Any cautions that should be exercised
* Notes for more advanced users
* References to the literature for more discussion or more examples, including pointers to electronic resources, such as URLs


https://www.amazon.com/dp/1584883553/?tag=pfamazon01-20
by A.D. Polyanin
Chapman & Hall/CRC (October 29, 2003)

The latest in a series of best-selling differential equation handbooks by these authors, the Handbook of Nonlinear Partial Differential Equations presents exact solutions of more than 1600 nonlinear equations encountered in science and engineering--many more than any other book available--including those of parabolic, hyperbolic, elliptic and mixed type. It describes a number of new solutions to nonlinear equations and pays special attention to equations of general form that involve arbitrary functions. A supplement at the end of the book discusses the basic methods for constructing exact solutions and outlines new direct methods of generalized and function separation of variables.

https://www.amazon.com/dp/0817643230/?tag=pfamazon01-20
by Lokenath Debnath
Birkhäuser Boston; 2 edition (December 17, 2004)

"An exceptionally complete overview. There are numerous examples and the emphasis is on applications to almost all areas of science and engineering. There is truly something for everyone here. This reviewer feels that it is a very hard act to follow, and recommends it strongly. [This book] is a jewel." ---Applied Mechanics Review (Review of First Edition)

This expanded and revised second edition is a comprehensive and systematic treatment of linear and nonlinear partial differential equations and their varied applications. Building upon the successful material of the first book, this edition contains updated modern examples and applications from areas of fluid dynamics, gas dynamics, plasma physics, nonlinear dynamics, quantum mechanics, nonlinear optics, acoustics, and wave propagation. Methods and properties of solutions are presented, along with their physical significance, making the book more useful for a diverse readership.

Nonlinear Partial Differential Equations for Scientists and Engineers, Second Edition is an exceptionally complete and accessible text/reference for graduate students, researchers, and professionals in mathematics, physics, and engineering. It may be used in graduate-level courses, as a self-study resource, or as a research reference.

Academic Press and CRC are reputable scientific/techical publishers

A.D. Polyanin has a number of texts in mathematics.

 

[phosgene]

Hi guys, holidays are over and I'm back to uni. Currently doing a quick summer course to catch up with one course I missed last year, we're currently doing differential equations. The textbook that I used for calculus last year has one section on DE, and it's not nearly enough. DE's only take up a little of the course I'm doing atm, we're only going to be doing first order seperable DE's, linear first order DE's and linear second order DE's with constant coefficients. However, right after the course ends, I will be taking an actual course on differential equations. So what I'm looking for is a textbook that will fulfill the needs of my current course, and also of the course I am about to take. Topics covered in the actual DE's course are: first order ordinary differential equations (ODEs), higher order ODEs, numerical techniques for solving ODEs, systems of ODEs, series solutions of ODEs, Laplace transforms, Fourier analysis, solution of linear partial differential equations using the method of separation of variables, and D'Alembert's solution of the wave equation.

So, in a nutshell, I'm looking for an introductory DE's textbook. Even just a short list of the best ones will do. There are just so many out there that it's kind of hard to pick one for myself.

 

[Greg Bernhardt]

• Author: Martin Braun
• Title: Differential Equations and Their Applications
• Amazon Link: https://www.amazon.com/dp/0387978941/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[LIST]
[*] First-order differential equations 
[LIST]
[*] Introduction
[*] First-order linear differential equations
[*] The Van Meegeren art forgeries
[*] Separable equations
[*] Population models
[*] The spread of technological innovations
[*] An atomic waste disposal problem
[*] The dynamics of tumor growth, mixing problems, and orthogonal trajectories
[*] Exact equations, and why we cannot solve very many differential equations
[*] The existence-uniqueness theorem; Picard iteration
[*] Finding roots of equations by iteration
[LIST]
[*] Newton's method
[/LIST]
[*] Difference equations, and how to compute the interest due on your student loans
[*] Numerical approximations; Euler's method
[LIST]
[*] Error analysis for Euler's method
[/LIST]
[*] The three term Taylor series method
[*] An improved Euler method
[*] The Runge-Kutta method
[*] What to do in practice
[/LIST]
[*] Second-order linear differential equations
[LIST] 
[*] Algebraic properties of solutions
[*] Linear equations with constant coefficients
[LIST]
[*] Complex roots
[*] Equal roots; reduction of order
[/LIST]
[*] The nonhomogeneous equation
[*] The method of variation of parameters
[*] The method of judicious guessing
[*] Mechanical vibrations
[LIST]
[*] The Tacoma Bridge disaster
[*] Electrical networks
[/LIST]
[*] A model for the detection of diabetes
[*] Series solutions
[LIST]
[*] Singular points; Euler equations
[*] Regular singular points; the method of Frobenius
[*] Equal roots, and roots differing by an integer
[/LIST]
[*] The method of Laplace transforms
[*] Some useful properties of Laplace transforms
[*] Differential equations with discontinuous right-hand sides
[*] The Dirac delta function
[*] The convolution integral
[*] The method of elimination for systems
[*] Higher-order equations
[/LIST]
[*] Systems of differential equations
[LIST]
[*] Algebraic properties of solutions of linear systems
[*] Vector spaces
[*] Dimension of a vector space
[*] Applications of linear algebra to differential equations
[*] The theory of determinants
[*] Solutions of simultaneous linear equations
[*] Linear transformations
[*] The eigenvalue-eigenvector method of finding solutions
[*] Complex roots
[*] Equal roots
[*] Fundamental matrix solutions; [itex]e^{At}[/itex]
[*] The nonhomogeneous equation; variation of parameters
[*] Solving systems by Laplace transforms
[/LIST]
[*] Qualitative theory of differential equations
[LIST]
[*] Introduction
[*] Stability of linear systems
[*] Stability of equilibrium solutions
[*] The phase-plane
[*] Mathematical theories of war
[LIST]
[*] L. F. Richardson's theory of conflict
[*] Lanchester's combat models and the battle of Iwo Jima
[/LIST]
[*] Qualitative properties of orbits
[*] Phase portraits of linear systems
[*] Long time behavior of solutions; the Poincare-Bendixson Theorem
[*] Introduction to bifurcation theory
[*] Predator-prey problems; or why the percentage of sharks caught in the Mediterranean Sea rose dramatically during World War I
[*] The principle of competitive exclusion in population biology
[*] The Threshold Theorem of epidemiology
[*] A model for the spread of gonorrhea
[/LIST]
[*] Separation of variables and Fourier series
[LIST]
[*] Two point boundary-value problems
[*] Introduction to partial differential equations
[*] The heat equation; separation of variables
[*] Fourier series
[*] Even and odd functions
[*] Return to the heat equation
[*] The wave equation
[*] Laplace's equation
[/LIST]
[*] Appendix: Some simple facts concerning functions of several variables
[*] Appendix: Sequences and series
[*] Appendix: Introduction to APL
[*] Answers to odd-numbered exercises
[*] Index
[/LIST]

 

[Greg Bernhardt]

• Author: Morris Tenenbaum
• Title: Ordinary Differential Equations
• Amazon Link: https://www.amazon.com/dp/0486649407/?tag=pfamazon01-20
• Prerequisities:

Table of Contents:

[LIST]
[*] Preface for the Teacher[*] Preface for the Student[*] Basic Concepts

[LIST]
[*] How Differential Equations Originate.
[*] The Meaning of the Terms Set and Function. Implicit Functions. Elementary Functions.
[LIST]
[*] The Meaning of the Term Set.
[*] The Meaning of the Term Function of One Independent Variable.
[*] Function of Two Independent Variables. 
[*] Implicit Function.
[*] The Elementary Functions.
[/LIST]
[*] The Differential Equation.
[LIST]
[*] Definition of an Ordinary Differential Equation. Order of a Differential Equation. 
[*] Solution of a Differential Equation. Explicit Solution. 
[*] Implicit Solution of a Differential Equation.
[/LIST]
[*] The General Solution of a Differential Equation.
[LIST]
[*] Multiplicity of Solutions of a Differential Equation. 
[*] Method of Finding a Differential Equation if Its [itex]n[/itex]-Parameter Family of Solutions Is Known. 
[*] General Solution. Particular Solution. Initial Conditions.
[/LIST]
[*] Direction Field.
[LIST] 
[*] Construction of a Direction Field. The Isoclines of a Direction Field. 
[*] The Ordinary and Singular Points of the First Order Equation (5.11).
[/LIST]
[/LIST][*] Special Types of Differential Equations of the First Order
[LIST]
[*] Meaning of the Differential of a Function. Separable Differential Equations.
[LIST]
[*] Differential of a Function of One Independent Variable.
[*] Differential of a Function of Two Independent Variables.
[*] Differential Equations with Separable Variables.
[/LIST]
[*] First Order Differential Equation with Homogeneous Coefficients.
[LIST]
[*] Definition of a Homogeneous Function. 
[*] Solution of a Differential Equation in Which the Coefficients of [itex]dx[/itex] and [itex]dy[/itex] Are Each Homogeneous Functions of the Same Order.
[/LIST]
[*] Differential Equations with Linear Coefficients.
[LIST]
[*] Review of Some Plane Analytic Geometry.
[*] Solution of a Differential Equation in Which the Coefficients of [itex]dx[/itex] and [itex]dy[/itex] are Linear, Nonhomogeneous, and When Equated to Zero Represent Non-parallel Lines.
[*] A Second Method of Solving the Differential Equation (8.2) with Nonhomogeneous Coefficients.
[*] Solution of a Differential Equation in Which the Coefficients of [itex]dx[/itex] and [itex]dy[/itex] Define Parallel or Coincident Lines.
[/LIST]
[*] Exact Differential Equations.
[LIST]
[*] Definition of an Exact Differential and of an Exact Differential Equation.
[*] Necessary and Sufficient Condition for Exactness and Method of Solving an Exact Differential Equation. 
[/LIST]
[*] Recognizable Exact Differential Equations. Integrating Factors.
[LIST]
[*] Recognizable Exact Differential Equations.
[*] Integrating Factors.
[*] Finding an Integrating Factor.
[/LIST]
[*] The Linear Differential Equation of the First Order. Bernoulli Equation.
[LIST]
[*] Definition of a Linear Differential Equation of the First Order.
[*] Method of Solution of a Linear Differential Equation of the First Order. 
[*] Determination of the Integrating Factor [itex]e^{\int P(x)dx}[/itex].
[*] Bernoulli Equation.
[/LIST]
[*] Miscellaneous Methods of Solving a First Order Differential Equation.
[LIST]
[*] Equations Permitting a Choice of Method.
[*] Solution by Substitution and Other Means
[/LIST]
[/LIST][*] Problems Leading to Differential Equations of the First Order
[LIST]
[*] Geometric Problems.
[*] Trajectories.
[LIST]
[*] Isogonal Trajectories. 
[*] Orthogonal Trajectories.
[*] Orthogonal Trajectory Formula in Polar Coordinates.
[/LIST]
[*] Dilution and Accretion Problems. Interest Problems. Temperature Problems. Decomposition and Growth Problems. Second Order Processes.
[LIST]
[*] Dilution and Accretion Problems.
[*] Interest Problems.
[*] Temperature Problems.
[*] Decomposition and Growth Problems.
[*] Second Order Processes.
[/LIST]
[*] Motion of a Particle Along a Straight Line — Vertical, Horizontal, Inclined.
[LIST]
[*] Vertical Motion.
[*] Horizontal Motion. 
[*] Inclined Motion.
[/LIST]
[*] Pursuit Curves. Relative Pursuit Curves.
[LIST]
[*] Pursuit Curves.
[*] Relative Pursuit Curve.
[/LIST]
[*] Miscellaneous Types of Problems Leading to Equations of the First Order
[LIST]
[*] Flow of Water Through an Orifice.
[*] First Order Linear Electric Circuit.
[*] Steady State Flow of Heat.
[*] Pressure—Atmospheric and Oceanic.
[*] Rope or Chain Around a Cylinder.
[*] Motion of a Complex System.
[*] Variable Mass. Rocket Motion.
[*] Rotation of the Liquid in a Cylinder.
[/LIST]
[/LIST][*] Linear Differential Equations of Order Greater Than One
[LIST]
[*] Complex Numbers and Complex Functions.
[LIST]
[*] Complex Numbers.
[*] Algebra of Complex Numbers.
[*] Exponential, Trigonometric, and Hyperbolic Functions of Complex Numbers.
[/LIST]
[*] Linear Independence of Functions. The Linear Differential Equation of Order [itex]n[/itex].
[LIST]
[*] Linear Independence of Functions.
[*] The Linear Differential Equation of Order [itex]n[/itex]
[/LIST]
[*] Solution of the Homogeneous Linear Differential Equation of Order [itex]n[/itex] with Constant Coefficients.
[LIST]
[*] General Form of Its Solutions.
[*] Roots of the Characteristic Equation (20.14) Real and Distinct.
[*] Roots of Characteristic Equation (20.14) Real but Some Multiple.
[*] Some or All Roots of the Characteristic Equation (20.14) Imaginary.
[/LIST]
[*] Solution of the Nonhomogeneous Linear Differential Equation of Order [itex]n[/itex] with Constant Coefficients.
[LIST]
[*] Solution by the Method of Undetermined Coefficients.
[*] Solution by the Use of Complex Variables.
[/LIST]
[*] Solution of the Nonhomogeneous Linear Differential Equation by the Method of Variation of Parameters.
[LIST]
[*] Introductory Remarks.
[*] The Method of Variation of Parameters.
[/LIST]
[*] Solution of the Linear Differential Equation with Nonconstant Coefficients. Reduction of Order Method.
[LIST]
[*] Introductory Remarks. 
[*] Solution of the Linear Differential Equation with Nonconstant Coefficients by the Reduction of Order Method.
[/LIST]
[/LIST][*] Operators and Laplace Transforms
[LIST]
[*] Differential and Polynomial Operators.
[LIST]
[*] Definition of an Operator. Linear Property of Polynomial Operators.
[*] Algebraic Properties of Polynomial Operators.
[*] Exponential Shift Theorem for Polynomial Operators.
[*] Solution of a Linear Differential Equation with Constant Coefficients by Means of Polynomial Operators.
[/LIST]
[*] Inverse Operators.
[LIST]
[*] Meaning of an Inverse Operator.
[*] Solution of (25.1) by Means of Inverse Operators.
[/LIST]
[*] Solution of a Linear Differential Equation by Means of the Partial Fraction Expansion of Inverse Operators.
[LIST] 
[*] Partial Fraction Expansion Theorem.
[*] First Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators.
[*] A Second Method of Solving a Linear Equation by Means of the Partial Fraction Expansion of Inverse Operators.
[/LIST]
[*] The Laplace Transform. Gamma Function.
[LIST]
[*] Improper Integral. Definition of a Laplace Transform.
[*] Properties of the Laplace Transform.
[*] Solution of a Linear Equation with Constant Coefficients by Means of a Laplace Transform.
[*] Construction of a Table of Laplace Transforms.
[*] The Gamma Function.
[/LIST]
[/LIST][*] Problems Leading to Linear Differential Equations of Order Two
[LIST] 
[*] Undamped Motion.
[LIST]
[*] Free Undamped Motion. (Simple Harmonic Motion.)
[*] Definitions in Connection with Simple Harmonic Motion.
[*] Examples of Particles Executing Simple Harmonic Motion. Harmonic Oscillators.
[*] Forced Undamped Motion.
[/LIST]
[*] Damped Motion.
[LIST]
[*] Free Damped Motion. (Damped Harmonic Motion.)
[*] Forced Motion with Damping.
[/LIST]
[*] Electric Circuits. Analog Computation.
[LIST]
[*] Simple Electric Circuit.
[*] Analog Computation.
[/LIST]
[*] Miscellaneous Types of Problems Leading to Linear Equations of the Second Order
[LIST]
[*] Problems Involving a Centrifugal Force.
[*] Rolling Bodies.
[*] Twisting Bodies.
[*] Bending of Beams.
[/LIST]
[/LIST][*] Systems of Differential Equations. Linearization of First Order Systems.
[LIST]
[*] Solution of a System of Differential Equations.
[LIST]
[*] Meaning of a Solution of a System of Differential Equations.
[*] Definition and Solution of a System of First Order Equations.
[*] Definition and Solution of a System of Linear First Order Equations.
[*] Solution of a System of Linear Equations with Constant Coefficients by the Use of Operators. Nondegenerate Case.
[*] An Equivalent Triangular System.
[*] Degenerate Case. [itex]f_1(D)g_2(D)-g_1(D)f_2(D)=0[/itex].
[*] Systems of Three Linear Equations.
[*] Solution of a System of Linear Differential Equations with Constant Coefficients by Means of Laplace Transforms.
[/LIST]
[*] Linearization of First Order Systems.
[/LIST][*] Problems Giving Rise to Systems of Equations. Special Types of Second Order Linear and Nonlinear Equations Solvable by Reducing to Systems.
[LIST]
[*] Mechanical, Biological, Electrical Problems Giving Rise to Systems of Equations.
[LIST]
[*] A Mechanical Problem — Coupled Springs. 
[*] A Biological Problem.
[*] An Electrical Problem. More Complex Circuits.
[/LIST]
[*] Plane Motions Giving Rise to Systems of Equations.
[LIST]
[*] Derivation of Velocity and Acceleration Formulas.
[*] The Plane Motion of a Projectile.
[*] Definition of a Central Force. Properties of the Motion of a Particle Subject to a Central Force.
[*] Definitions of Force Field, Potential, Conservative Field. Conservation of Energy in a Conservative Field.
[*] Path of a Particle in Motion Subject to a Central Force Whose Magnitude Is Proportional 
to Its Distance from a Fixed Point O. 
[*] Path of a Particle in Motion Subject to a Central Force Whose Magnitude Is Inversely Proportional to the Square of Its Distance from a Fixed Point O.
[*] Planetary Motion. 
[*] H. Kepler's (1571-1630) Laws of Planetary Motion. Proof of Newton's Inverse Square Law.
[/LIST]
[*] Special Types of Second Order Linear and Nonlinear Differential Equations Solvable by Reduction to a System of Two First Order Equations.
[LIST]
[*] Solution of a Second Order Nonlinear Differential Equation in Which [itex]y^\prime[/itex] and the Independent Variable [itex]x[/itex] Are Absent.
[*] Solution of a Second Order Nonlinear Differential Equation in Which the Dependent Variable [itex]y[/itex] Is Absent.
[*] Solution of a Second Order Nonlinear Equation in Which the Independent Variable [itex]x[/itex] Is Absent.
[/LIST]
[*] Problems Giving Rise to Special Types of Second Order Nonlinear Equations.
[LIST]
[*] The Suspension Cable.
[*] A Special Central Force Problem.
[*] A Pursuit Problem Leading to a Second Order Nonlinear Differential Equation.
[*] Geometric Problems.
[/LIST]
[/LIST][*] Series Methods
[LIST]
[*] Power Series Solutions of Linear Differential Equations.
[LIST]
[*] Review of Taylor Series and Related Matters.
[*] Solution of Linear Differential Equations by Series Methods.
[/LIST]
[*] Series Solution of [itex]y^\prime = f(x,y)[/itex].
[*] Series Solution of a Nonlinear Differential Equation of Order Greater Than One and of a System of First Order Differential Equations.
[LIST]
[*] Series Solution of a System of First Order Differential Equations.
[*] Series Solution of a System of Linear First Order Equations.
[*] Series Solution of a Nonlinear Differential Equation of Order Greater Than One.
[/LIST]
[*] Ordinary Points and Singularities of a Linear Differential Equation. Method of Frobenius. 
[LIST]
[*] Ordinary Points and Singularities of a Linear Differential Equation.
[*] Solution of a Homogeneous Linear Differential Equation About a Regular Singularity. Method of Frobenius.
[/LIST]
[*] The Legendre Differential Equation. Legendre Functions. Legendre Polynomials [itex]P_k(x)[/itex]. Properties of Legendre Polynomials [itex]P_k(x)[/itex].
[LIST]
[*] The Legendre Differential Equation.
[*] Comments on the Solution (41.18) of the Legendre Equation (41.1). Legendre Functions. 
Legendre Polynomials [itex]P_k(x)[/itex].
[*] Properties of Legendre Polynomials [itex]P_k(x)[/itex]
[/LIST]
[*] The Bessel Differential Equation. Bessel Function of the First Kind [itex]J_k(x)[/itex], Differential Equations Leading to a Bessel Equation. Properties of [itex]J_k(x)[/itex]
[LIST]
[*] The Bessel Differential Equation.
[*] Bessel Functions of the First Kind [itex]J_k(x)[/itex].
[*] Differential Equations Which Lead to a Bessel Equation.
[*] Properties of Bessel Functions of the First Kind [itex]J_k(x)[/itex]
[/LIST]
[*] The Laguerre Differential Equation. Laguerre Polynomials [itex]L_k(x)[/itex]. Properties of [itex]L_k(x)[/itex]
[LIST]
[*] The Laguerre Differential Equation and Its Solution.
[*] The Laguerre Polynomial [itex]L_k(x)[/itex]. 
[*] Some Properties of Laguerre Polynomials [itex]L_k(x)[/itex]
[/LIST]
[/LIST][*] Numerical Methods
[LIST]
[*] Starting Method. Polygonal Approximation.
[*] An Improvement of the Polygonal Starting Method.
[*] Starting Method — Taylor Series.
[LIST]
[*] Numerical Solution of [itex]y^\prime = f(x,y)[/itex] by Direct Substitution in a Taylor Series.
[*] Numerical Solution of [itex]y^\prime = f(x,y)[/itex] by the "Creeping Up" Process.
[/LIST]
[*] Starting Method — Runge-Kutta Formulas.
[*] Finite Differences. Interpolation.
[LIST]
[*] Finite Differences.
[*] Polynomial Interpolation.
[/LIST]
[*] Newton's Interpolation Formulas.
[LIST]
[*] Newton's (Forward) Interpolation Formula.
[*] Newton's (Backward) Interpolation Formula.
[*] The Error in Polynomial Interpolation.
[/LIST]
[*] Approximation Formulas Including Simpson's and Weddle's Rule.
[*] Milne's Method of Finding an Approximate Numerical Solution of [itex]y' = f(x,y)[/itex].
[*] General Comments. Selecting [itex]h[/itex]. Reducing [itex]h[/itex]. Summary and an Example.
[LIST]
[*] Comment on Errors.
[*] Choosing the Size of [itex]h[/itex].
[*] Reducing and Increasing [itex]h[/itex].
[*] Summary and an Illustrative Example.
[/LIST]
[*] Numerical Methods Applied to a System of Two First Order Equations.
[*] Numerical Solution of a Second Order Differential Equation.
[*] Perturbation Method. First Order Equation.
[*] Perturbation Method. Second Order Equation.
[/LIST][*] Existence and Uniqueness Theorem for the First Order Differential Equation [itex]y^\prime= f(x,y)[/itex]. Picard's Method. Envelopes. Clairaut Equation.
[LIST]
[*] Picard's Method of Successive Approximations.
[*] An Existence and Uniqueness Theorem for the First Order Differential Equation [itex]y^\prime = f(x,y)[/itex] Satisfying [itex]y(x_0)=y_0[/itex].
[LIST]
[*] Convergence and Uniform Convergence of a Sequence of Functions. Definition of a Continuous Function.
[*] Lipschitz Condition. Theorems from Analysis.
[*] Proof of the Existence and Uniqueness Theorem for the First Order Differential Equation [itex]y^\prime = f(x,y)[/itex].
[/LIST]
[*] The Ordinary and Singular Points of a First Order Differential Equation [itex]y^\prime = f(x,y)[/itex].
[*] Envelopes.
[LIST]
[*] Envelopes of a Family of Curves.
[*] Envelopes of a 1-Parameter Family of Solutions.
[/LIST]
[*] The Clairaut Equation.
[/LIST][*] Existence and Uniqueness Theorems for a System of First Order Differential equations and for Linear and Nonlinear Differential Equations of Order Greater Than One. Wronskians.
[LIST]
[*] An Existence and Uniqueness Theorem for a System of [itex]n[/itex] First Order Differential Equations and for a Nonlinear Differential Equation of Order Greater Than One.
[LIST]
[*] The Existence and Uniqueness Theorem for a System of [itex]n[/itex] First Order Differential Equations.
[*] Existence and Uniqueness Theorem for a Nonlinear Differential Equation of Order [itex]n[/itex]. 
[*] Existence and Uniqueness Theorem for a System of [itex]n[/itex] Linear First Order 
Equations.
[/LIST]
[*] Determinants. Wronskians.
[LIST]
[*] A Brief Introduction to the Theory of Determinants.
[*] Wronskians.
[/LIST]
[*] Theorems About Wronskians and the Linear Independence of a Set of Solutions of a Homogeneous Linear Differential Equation.
[*] Existence and Uniqueness Theorem for the Linear Differential Equation of Order [itex]n[/itex].
[/LIST][*] Bibliography[*] Index
[/LIST]

 

[Astronuc]

• Author: George Simmons, Steven Krantz
• Title: Differential Equations: Theory, Technique, and Practice (Walter Rudin Student Series in Advanced Mathematics)
• Amazon Link: https://www.amazon.com/dp/0072863153/?tag=pfamazon01-20
• Prerequisities: Introductory Calculus
• Level: Undergraduate, Introductory

Publisher's site: http://highered.mcgraw-hill.com/sites/0072863153/

Table of Contents:

Preface 
1 What is a Differential Equation?
1.1 Introductory Remarks 
1.2 The Nature of Solutions 
1.3 Separable Equations 
1.4 First-Order Linear Equations 
1.5 Exact Equations 
1.6 Orthogonal Trajectories and Families of Curves 
1.7 Homogeneous Equations 
1.8 Integrating Factors 
1.9 Reduction of Order 
1.9.1 Dependent Variable Missing 
1.9.2 Independent Variable Missing 
1.10 The Hanging Chain and Pursuit Curves 
1.10.1 The Hanging Chain 
1.10.2 Pursuit Curves 
1.11 Electrical Circuits 
Anatomy of an Application: The Design of a Dialysis Machine 
Problems for Review and Discovery 


2 Second-Order Linear Equations 
2.1 Second-Order Linear Equations with Constant Coefficients 
2.2 The Method of Undetermined Coefficients 
2.3 The Method of Variation of Parameters 
2.4 The Use of a Known Solution to Find Another 
2.5 Vibrations and Oscillations 
2.5.1 Undamped Simple Harmonic Motion 
2.5.2 Damped Vibrations 
2.5.3 Forced Vibrations 
2.5.4 A Few Remarks About Electricity 
2.6 Newton’s Law of Gravitation and Kepler’s Laws 
2.6.1 Kepler’s Second Law 
2.6.2 Kepler’s First Law 
2.6.3 Kepler’s Third Law 
2.7 Higher Order Linear Equations, Coupled Harmonic Oscillators 
Historical Note: Euler 
Anatomy of an Application: Bessel Functions and the Vibrating Membrane 
Problems for Review and Discovery 


3 Qualitative Properties and Theoretical Aspects 
3.1 Review of Linear Algebra 
3.1.1 Vector Spaces 
3.1.2 The Concept Linear Independence 
3.1.3 Bases 
3.1.4 Inner Product Spaces 
3.1.5 Linear Transformations and Matrices 
3.1.6 Eigenvalues and Eigenvectors 
3.2 A Bit of Theory 
3.3 Picard’s Existence and Uniqueness Theorem 
3.3.1 The Form of a Differential Equation 
3.3.2 Picard’s Iteration Technique 
3.3.3 Some Illustrative Examples 
3.3.4 Estimation of the Picard Iterates 
3.4 Oscillations and the Sturm Separation Theorem 
3.5 The Sturm Comparison Theorem 
Anatomy of an Application: The Green’s Function 
Problems for Review and Discovery 


4 Power Series Solutions and Special Functions 
4.1 Introduction and Review of Power Series 
4.1.1 Review of Power Series 
4.2 Series Solutions of First-Order Differential Equations 
4.3 Second-Order Linear Equations: Ordinary Points 
4.4 Regular Singular Points 
4.5 More on Regular Singular Points 
4.6 Gauss’s Hypergeometric Equation 
Historical Note: Gauss 
Historical Note: Abel 
Anatomy of an Application: Steady-State Temperature in a Ball 
Problems for Review and Discovery 


5 Fourier Series: Basic Concepts 
5.1 Fourier Coefficients 
5.2 Some Remarks about Convergence 
5.3 Even and Odd Functions: Cosine and Sine Series 
5.4 Fourier Series on Arbitrary Intervals 
5.5 Orthogonal Functions 
Historical Note: Riemann 
Anatomy of an Application: Introduction to the Fourier Transform 
Problems for Review and Discovery 


6 Partial Differential Equations and Boundary Value Problems 
6.1 Introduction and Historical Remarks 
6.2 Eigenvalues, Eigenfunctions, and the Vibrating String 
6.2.1 Boundary Value Problems 
6.2.2 Derivation of the Wave Equation 
6.2.3 Solution of the Wave Equation 
6.3 The Heat Equation 
6.4 The Dirichlet Problem for a Disc 
6.4.1 The Poisson Integral 
6.5 Sturm-Liouville Problems 
Historical Note: Fourier 
Historical Note: Dirichlet 
Anatomy of an Application: Some Ideas from Quantum Mechanics 
Problems for Review and Discovery 


7 Laplace Transforms 
7.1 Introduction 
7.2 Applications to Differential Equations 
7.3 Derivatives and Integrals of Laplace Transforms 
7.4 Convolutions 
7.3.1 Abel's Mechanical Problem 
7.5 The Unit Step and Impulse Functions 
Historical Note: Laplace 
Anatomy of an Application: Flow Initiated by an Impulsively-Started Flat Plate 
Problems for Review and Discovery 


8 The Calculus of Variations 
8.1 Introductory Remarks 
8.2 Euler’s Equation 
8.3 Isoperimetric Problems and the Like 
8.3.1 Lagrange Multipliers 
8.3.2 Integral Side Conditions 
8.3.3 Finite Side Conditions 
Historical Note: Newton 
Anatomy of an Application: Hamilton’s Principle and its Implications 
Problems for Review and Discovery 


9 Numerical Methods 
9.1 Introductory Remarks 
9.2 The Method of Euler 
9.3 The Error Term 
9.4 An Improved Euler Method 
9.5 The Runge-Kutta Method 
Anatomy of an Application: A Constant Perturbation Method for Linear, Second-Order Equations 
Problems for Review and Discovery 


10 Systems of First-Order Equations 
10.1 Introductory Remarks 
10.2 Linear Systems 
10.3 Homogeneous Linear Systems with Constant Coefficients 
10.4 Nonlinear Systems: Volterra’s Predator-Prey Equations 
Anatomy of an Application: Solution of Systems with Matrices and Exponentials 
Problems for Review and Discovery 


11 The Nonlinear Theory 
11.1 Some Motivating Examples 
11.2 Specializing Down 
11.3 Types of Critical Points: Stability 
11.4 Critical Points and Stability for Linear Systems 
11.5 Stability by Liapunov’s Direct Method 
11.6 Simple Critical Points of Nonlinear Systems 
11.7 Nonlinear Mechanics: Conservative Systems 
11.8 Periodic Solutions: The Poincaré-Bendixson Theorem 
Historical Note: Poincaré 
Anatomy of an Application: Mechanical Analysis of a Block on a Spring 
Problems for Review and Discovery 


12 Dynamical Systems 
12.1 Flows 
12.1.1 Dynamical Systems 
12.1.2 Stable and Unstable Fixed Points 
12.1.3 Linear Dynamics in the Plane 
12.2 Some Ideas from Topology 
12.2.1 Open and Closed Sets 
12.2.2 The Idea of Connectedness 
12.2.3 Closed Curves in the Plane 
12.3 Planar Autonomous Systems 
12.3.1 Ingredients of the Proof of Poincaré-Bendixson 
Anatomy of an Application: Lagrange’s Equations 
Problems for Review and Discovery 

Bibliography

I used, and still have, the 1972 edition of George Simmons's, Differential Equations with Applications and Historical Notes, which was one of the books in McGraw-Hill's International Series in Pure and Applied Mathematics. The text was revised in 1991.

1991 Ed - https://www.amazon.com/dp/0070575401/?tag=pfamazon01-20

Nathaniel Grossman, Professor of Mathematics, UCLA, writes:

As with his excellent calculus textbook, the author tries to show students how mathematics is a human activity, a subject that developed in response to actual needs and which is still lively and developing. No part of mathematics illustrates this development better than the topic of differential equations, which was invented to solve pressing problems in astronomy. One example: In Newton's time, accurate location of position on the open seas was an unsolved problem, crucial to commerce. New techniques from differential equations led to the ready calculation of tables which, together with the invention of Harrison's sea-going chronometer, effectively solved the navigation problem. Differential equations lie at the core of the physical sciences and engineering and are proving increasingly valuable in biology and medicine.

 

[Astronuc]

• Author: Gerald B. Folland
• Title: Introduction to Partial Differential Equations
• Amazon Link: https://www.amazon.com/dp/0691043612/?tag=pfamazon01-20
• Prerequisities: Calculus (including multivariable), Differential equations,
• Level: Undergraduate, intermediate

Table of Contents

Local Existence Theory
The Laplace Operator
Layer Potentials
The Heat Operator
The Wave Operator
The L2 Theory of Derivatives
Elliptic Boundary Value Problems
Pseudodifferential Operators

The second edition of Introduction to Partial Differential Equations, which originally appeared in the Princeton series Mathematical Notes, serves as a text for mathematics students at the intermediate graduate level. The goal is to acquaint readers with the fundamental classical results of partial differential equations and to guide them into some aspects of the modern theory to the point where they will be equipped to read advanced treatises and research papers. This book includes many more exercises than the first edition, offers a new chapter on pseudodifferential operators, and contains additional material throughout.

The first five chapters of the book deal with classical theory: first-order equations, local existence theorems, and an extensive discussion of the fundamental differential equations of mathematical physics. The techniques of modern analysis, such as distributions and Hilbert spaces, are used wherever appropriate to illuminate these long-studied topics. The last three chapters introduce the modern theory: Sobolev spaces, elliptic boundary value problems, and pseudodifferential operators.

http://press.princeton.edu/titles/5860.html

 

[Astronuc]

• Author: Lawrence C. Evans
• Title: Partial Differential Equations
• Amazon Link: https://www.amazon.com/dp/0821849743/?tag=pfamazon01-20
• Prerequisities: Undergraduate degree in mathematics and/or physics, engineering
• Level: Graduate

Table of Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1
1.1. Partial differential equations . . . . . . . . . . . . 1
1.2. Examples . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1. Single partial differential equations . . . . . . . .3
1.2.2. Systems of partial differential equations . . . . . .6
1.3. Strategies for studying PDE . . . . . . . . . . . . . .6
1.3.1. Well-posed problems, classical solutions . . . . . . 7
1.3.2. Weak solutions and regularity . . . . . . . . . . . .7
1.3.3. Typical difficulties . . . . . . . . . . . . . . . . 9
1.4. Overview . . . . . . . . . . . . . . . . . . . . . . . 9
1.5. Problems . . . . . . . . . . . . . . . . . . . . . . .12
1.6. References . . . . . . . . . . . . . . . . . . . . . .13

PART I: REPRESENTATION FORMULAS FOR SOLUTIONS
2. Four Important Linear PDE . . . . . . . . . . . . . . . 17
2.1. Transport equation . . . . . . . . . . . . . . . . . .18
2.1.1. Initial-value problem . . . . . . . . . . . . . . . 18
2.1.2. Nonhomogeneous problem . . . . . . . . . . . . . . .19
2.2. Laplace’s equation . . . . . . . . . . . . . . . . . .20
2.2.1. Fundamental solution . . . . . . . . . . . . . . . .21
2.2.2. Mean-value formulas . . . . . . . . . . . . . . . . 25
2.2.3. Properties of harmonic functions . . . . . . . . . .26
2.2.4. Green’s function . . . . . . . . . . . . . . . . . .33
2.2.5. Energy methods . . . . . . . . . . . . . . . . . . .41
2.3. Heat equation . . . . . . . . . . . . . . . . . . . . 44
2.3.1. Fundamental solution . . . . . . . . . . . . . . . .45
2.3.2. Mean-value formula . . . . . . . . . . . . . . . . .51
2.3.3. Properties of solutions . . . . . . . . . . . . . . 55
2.3.4. Energy methods . . . . . . . . . . . . . . . . . . .62
2.4. Wave equation . . . . . . . . . . . . . . . . . . . . 65
2.4.1. Solution by spherical means . . . . . . . . . . . . 67
2.4.2. Nonhomogeneous problem . . . . . . . . . . . . . . .80
2.4.3. Energy methods . . . . . . . . . . . . . . . . . . .82
2.5. Problems . . . . . . . . . . . . . . . . . . . . . . .84
2.6. References . . . . . . . . . . . . . . . . . . . . . .90

3. Nonlinear First-Order PDE. . . . . . . . . . . . . . . .91
3.1. Complete integrals, envelopes . . . . . . . . . . . . 92
3.1.1. Complete integrals . . . . . . . . . . . . . . . . .92
3.1.2. New solutions from envelopes . . . . . . . . . . . .94
3.2. Characteristics . . . . . . . . . . . . . . . . . . . 96
3.2.1. Derivation of characteristic ODE . . . . . . . . . .96
3.2.2. Examples . . . . . . . . . . . . . . . . . . . . . .99
3.2.3. Boundary conditions . . . . . . . . . . . . . . . .102
3.2.4. Local solution . . . . . . . . . . . . . . . . . . 105
3.2.5. Applications . . . . . . . . . . . . . . . . . . . 109
3.3. Introduction to Hamilton–Jacobi equations . . . . . .114
3.3.1. Calculus of variations, Hamilton’s ODE . . . . . . 115
3.3.2. Legendre transform, Hopf–Lax formula . . . . . . . 120
3.3.3. Weak solutions, uniqueness . . . . . . . . . . . . 128
3.4. Introduction to conservation laws . . . . . . . . . .135
3.4.1. Shocks, entropy condition . . . . . . . . . . . . .136
3.4.2. Lax–Oleinik formula . . . . . . . . . . . . . . . .143
3.4.3. Weak solutions, uniqueness . . . . . . . . . . . . 148
3.4.4. Riemann’s problem . . . . . . . . . . . . . . . . .153
3.4.5. Long time behavior . . . . . . . . . . . . . . . . 156
3.5. Problems . . . . . . . . . . . . . . . . . . . . . . 161
3.6. References . . . . . . . . . . . . . . . . . . . . . 165

4. Other Ways to Represent Solutions . . . . . . . . . . .167
4.1. Separation of variables . . . . . . . . . . . . . .  167
4.1.1. Examples . . . . . . . . . . . . . . . . . . . . . 168
4.1.2. Application: Turing instability . . . . . . . . . .172
4.2. Similarity solutions . . . . . . . . . . . . . . . . 176
4.2.1. Plane and traveling waves, solitons . . . . . . . .176
4.2.2. Similarity under scaling . . . . . . . . . . . . . 185
4.3. Transform methods . . . . . . . . . . . . . . . . . .187
4.3.1. Fourier transform . . . . . . . . . . . . . . . . .187
4.3.2. Radon transform . . . . . . . . . . . . . . . . . .196
4.3.3. Laplace transform . . . . . . . . . . . . . . . . .203
4.4. Converting nonlinear into linear PDE . . . . . . . . 206
4.4.1. Cole–Hopf transformation . . . . . . . . . . . . . 206
4.4.2. Potential functions . . . . . . . . . . . . . . . .208
4.4.3. Hodograph and Legendre transforms . . . . . . . . .209
4.5. Asymptotics . . . . . . . . . . . . . . . . . . . . .211
4.5.1. Singular perturbations . . . . . . . . . . . . . . 211
4.5.2. Laplace’s method . . . . . . . . . . . . . . . . . 216
4.5.3. Geometric optics, stationary phase . . . . . . . . 218
4.5.4. Homogenization . . . . . . . . . . . . . . . . . . 229
4.6. Power series . . . . . . . . . . . . . . . . . . . . 232
4.6.1. Noncharacteristic surfaces . . . . . . . . . . . . 232
4.6.2. Real analytic functions . . . . . . . . . . . . . .237
4.6.3. Cauchy–Kovalevskaya Theorem . . . . . . . . . . . .239
4.7. Problems . . . . . . . . . . . . . . . . . . . . . . 244
4.8. References . . . . . . . . . . . . . . . . . . . . . 249

PART II: THEORY FOR LINEAR PARTIAL DIFFERENTIAL EQUATIONS

5. Sobolev Spaces . . . . . . . . . . . . . . . . . . . . 253
5.1. H¨older spaces . . . . . . . . . . . . . . . . . . . 254
5.2. Sobolev spaces . . . . . . . . . . . . . . . . . . . 255
5.2.1. Weak derivatives . . . . . . . . . . . . . . . . . 255
5.2.2. Definition of Sobolev spaces . . . . . . . . . . . 258
5.2.3. Elementary properties . . . . . . . . . . . . . . .261
5.3. Approximation . . . . . . . . . . . . . . . . . . . .264
5.3.1. Interior approximation by smooth functions . . . . 264
5.3.2. Approximation by smooth functions . . . . . . . . .265
5.3.3. Global approximation by smooth functions . . . . . 266
5.4. Extensions . . . . . . . . . . . . . . . . . . . . . 268
5.5. Traces . . . . . . . . . . . . . . . . . . . . . . . 271
5.6. Sobolev inequalities . . . . . . . . . . . . . . . . 275
5.6.1. Gagliardo–Nirenberg–Sobolev inequality . . . . . . 276
5.6.2. Morrey’s inequality . . . . . . . . . . . . . . . .280
5.6.3. General Sobolev inequalities . . . . . . . . . . . 284
5.7. Compactness . . . . . . . . . . . . . . . . . . . . .286
5.8. Additional topics . . . . . . . . . . . . . . . . . .289
5.8.1. Poincar´e’s inequalities . . . . . . . . . . . . . 289
5.8.2. Difference quotients . . . . . . . . . . . . . . . 291
5.8.3. Differentiability a.e. . . . . . . . . . . . . . . 295
5.8.4. Hardy’s inequality . . . . . . . . . . . . . . . . 296
5.8.5. Fourier transform methods . . . . . . . . . . . . .297
5.9. Other spaces of functions . . . . . . . . . . . . . .299
5.9.1. The space H-1 . . . . . . . . . . . . . . . . . . .299
5.9.2. Spaces involving time . . . . . . . . . . . . . . .301
5.10. Problems . . . . . . . . . . . . . . . . . . . . . .305
5.11. References . . . . . . . . . . . . . . . . . . . . .309

6. Second-Order Elliptic Equations . . . . . . . . . . . .311
6.1. Definitions . . . . . . . . . . . . . . . . . . . . .311
6.1.1. Elliptic equations . . . . . . . . . . . . . . . . 311
6.1.2. Weak solutions . . . . . . . . . . . . . . . . . . 313
6.2. Existence of weak solutions . . . . . . . . . . . . .315
6.2.1. Lax–Milgram Theorem . . . . . . . . . . . . . . . .315
6.2.2. Energy estimates . . . . . . . . . . . . . . . . . 317
6.2.3. Fredholm alternative . . . . . . . . . . . . . . . 320
6.3. Regularity . . . . . . . . . . . . . . . . . . . . . 326
6.3.1. Interior regularity . . . . . . . . . . . . . . . .327
6.3.2. Boundary regularity . . . . . . . . . . . . . . . .334
6.4. Maximum principles . . . . . . . . . . . . . . . . . 344
6.4.1. Weak maximum principle . . . . . . . . . . . . . . 344
6.4.2. Strong maximum principle . . . . . . . . . . . . . 347
6.4.3. Harnack’s inequality . . . . . . . . . . . . . . . 351
6.5. Eigenvalues and eigenfunctions . . . . . . . . . . . 354
6.5.1. Eigenvalues of symmetric elliptic operators . . . .354
6.5.2. Eigenvalues of nonsymmetric elliptic operators . . 360
6.6. Problems . . . . . . . . . . . . . . . . . . . . . . 365
6.7. References . . . . . . . . . . . . . . . . . . . . . 370

7. Linear Evolution Equations . . . . . . . . . . . . . . 371
7.1. Second-order parabolic equations . . . . . . . . . . 371
7.1.1. Definitions . . . . . . . . . . . . . . . . . . . .372
7.1.2. Existence of weak solutions . . . . . . . . . . . .375
7.1.3. Regularity . . . . . . . . . . . . . . . . . . . . 380
7.1.4. Maximum principles . . . . . . . . . . . . . . . . 389
7.2. Second-order hyperbolic equations . . . . . . . . . .398
7.2.1. Definitions . . . . . . . . . . . . . . . . . . . .398
7.2.2. Existence of weak solutions . . . . . . . . . . . .401
7.2.3. Regularity . . . . . . . . . . . . . . . . . . . . 408
7.2.4. Propagation of disturbances . . . . . . . . . . . .414
7.2.5. Equations in two variables . . . . . . . . . . . . 418
7.3. Hyperbolic systems of first-order equations . . . . .421
7.3.1. Definitions . . . . . . . . . . . . . . . . . . . .421
7.3.2. Symmetric hyperbolic systems . . . . . . . . . . . 423
7.3.3. Systems with constant coefficients . . . . . . . . 429
7.4. Semigroup theory . . . . . . . . . . . . . . . . . . 433
7.4.1. Definitions, elementary properties . . . . . . . . 434
7.4.2. Generating contraction semigroups . . . . . . . . .439
7.4.3. Applications . . . . . . . . . . . . . . . . . . . 441
7.5. Problems . . . . . . . . . . . . . . . . . . . . . . 446
7.6. References . . . . . . . . . . . . . . . . . . . . . 449

Preview material - http://www.ams.org/bookstore/pspdf/gsm-19-r-prev.pdf
http://www.ams.org/bookstore-getitem/item=gsm-19-R

Readership: Graduate students and research mathematicians interested in partial differential equations.