Dennis Zill, Ph.D., received a doctorate in applied mathematics from Iowa State University and is a former professor of mathematics at Loyola Marymount University in Los Angeles, Loras College in Iowa and California Polytechnic State University. He is also the former chair of the mathematics department at Loyola Marymount University, where he currently holds the title of Professor Emeritus of Mathematics. Zill has interests in astronomy, modern literature, music, golf and good wine, while his research interests include special functions, differential equations, integral transformations and complex analysis.

• 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 Equations. Linear Equations. Exact Equations. 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 DEs. Chapter 3 in Review.4.HIGHER-ORDER DIFFERENTIAL EQUATIONS.Theory of 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. Green's Functions. Solving Systems of Linear DEs 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.THE 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 DIFFERENTIAL EQUATIONS.Theory of Linear Systems. Homogeneous Linear Systems. Nonhomogeneous Linear Systems. Matrix Exponential. Chapter 8 in Review.9.NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS. Euler Methods and Error Analysis. Runge-Kutta Methods. Multistep Methods. Higher-Order Equations and Systems. Second-Order Boundary-Value Problems. Chapter 9 in Review.10.SYSTEMS OF NONLINEAR DIFFERENTIAL EQUATIONS.Autonomous Systems. Stability of Linear Systems. Linearization and Local Stability. Autonomous Systems as Mathematical Models. Chapter 10 in Review.11.FOURIER SERIES.Orthogonal Functions. Fourier Series. 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 PDEs 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. Finite 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 A: Integral-Defined Functions.Appendix B: Matrices.Appendix C: Table of Laplace Transforms. Answers to Selected Odd-Numbered Problems. Index.