DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 8th Edition strikes a balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. This proven and accessible book speaks to beginning engineering and math students through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, definitions, and group projects. Written in a straightforward, readable, and helpful style, the book provides a thorough treatment of boundary-value problems and partial differential equations.
Table of Contents
1. INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminology. Initial-Value Problems. Differential Equations as Mathematical Models. Chapter 1 in Review. 2. FIRST-ORDER DIFFERENTIAL EQUATIONS. Solution
Curves Without a Solution. Separable Variables. Linear Equations. Exact
Equations and Integrating Factors. Solutions by Substitutions. A
Numerical Method. Chapter 2 in Review. 3. MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS. Linear Models. Nonlinear Models. Modeling with Systems of First-Order Differential Equations. Chapter 3 in Review. 4. HIGHER-ORDER DIFFERENTIAL EQUATIONS. Preliminary
Theory-Linear Equations. Reduction of Order. Homogeneous Linear
Equations with Constant Coefficients. Undetermined
Coefficients-Superposition Approach. Undetermined
Coefficients-Annihilator Approach. Variation of Parameters.
Cauchy-Euler Equation. Solving Systems of Linear Differential Equations
by Elimination. Nonlinear Differential Equations. Chapter 4 in Review. 5. MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS. Linear Models: Initial-Value Problems. Linear Models: Boundary-Value Problems. Nonlinear Models. Chapter 5 in Review. 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Review of Power Series Solutions About Ordinary Points. Solutions About Singular Points. Special Functions. Chapter 6 in Review. 7. LAPLACE TRANSFORM. Definition
of the Laplace Transform. Inverse Transform and Transforms of
Derivatives. Operational Properties I. Operational Properties II. Dirac
Delta Function. Systems of Linear Differential Equations. Chapter 7 in
Review. 8. SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS. Preliminary Theory. Homogeneous Linear Systems. Nonhomogeneous Linear Systems. Matrix Exponential. Chapter 8 in Review. 9. NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS. Euler
Methods. Runge-Kutta Methods. Multistep Methods. Higher-Order Equations
and Systems. Second-Order Boundary-Value Problems. Chapter 9 in Review. 10. PLANE AUTONOMOUS SYSTEMS. Autonomous
Systems. Stability of Linear Systems. Linearization and Local
Stability. Autonomous Systems as Mathematical Models. Chapter 10 in
Review. 11. ORTHOGONAL FUNCTIONS AND FOURIER SERIES. Orthogonal
Functions. Fourier Series and Orthogonal Functions. Fourier Cosine and
Sine Series. Sturm-Liouville Problem. Bessel and Legendre Series.
Chapter 11 in Review. 12. BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES. Separable
Partial Differential Equations. Classical PDE's and Boundary-Value
Problems. Heat Equation. Wave Equation. Laplace's Equation.
Nonhomogeneous Boundary-Value Problems. Orthogonal Series Expansions.
Higher-Dimensional Problems. Chapter 12 in Review. 13. BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS. Polar Coordinates. Polar and Cylindrical Coordinates. Spherical Coordinates. Chapter 13 in Review. 14. INTEGRAL TRANSFORM METHOD. Error Function. Laplace Transform. Fourier Integral. Fourier Transforms. Chapter 14 in Review. 15. NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS. Laplace's Equation. Heat Equation. Wave Equation. Chapter 15 in Review. Appendix I: Gamma Function. Appendix II: Matrices. Appendix III: Laplace Transforms. Answers for Selected Odd-Numbered Problems.