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1. INTRODUCTION TO DIFFERENTIAL EQUATIONS. Definitions and Terminology. InitialValue Problems. Differential Equations as Mathematical Models. Chapter 1 in Review. 2. FIRSTORDER 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 FIRSTORDER DIFFERENTIAL EQUATIONS. Linear Models. Nonlinear Models. Modeling with Systems of FirstOrder Differential Equations. Chapter 3 in Review. 4. HIGHERORDER DIFFERENTIAL EQUATIONS. Preliminary TheoryLinear Equations. Reduction of Order. Homogeneous Linear Equations with Constant Coefficients. Undetermined CoefficientsSuperposition Approach. Undetermined CoefficientsAnnihilator Approach. Variation of Parameters. CauchyEuler Equation. Solving Systems of Linear Differential Equations by Elimination. Nonlinear Differential Equations. Chapter 4 in Review. 5. MODELING WITH HIGHERORDER DIFFERENTIAL EQUATIONS. Linear Models: InitialValue Problems. Linear Models: BoundaryValue 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 FIRSTORDER 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. RungeKutta Methods. Multistep Methods. HigherOrder Equations and Systems. SecondOrder BoundaryValue 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. SturmLiouville Problem. Bessel and Legendre Series. Chapter 11 in Review. 12. BOUNDARYVALUE PROBLEMS IN RECTANGULAR COORDINATES. Separable Partial Differential Equations. Classical PDE's and BoundaryValue Problems. Heat Equation. Wave Equation. Laplace's Equation. Nonhomogeneous BoundaryValue Problems. Orthogonal Series Expansions. HigherDimensional Problems. Chapter 12 in Review. 13. BOUNDARYVALUE 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 OddNumbered Problems.
Dennis G. Zill is professor of mathematics at Loyola Marymount University. His interests are in applied mathematics, special functions, and integral transforms. Dr. Zill received his Ph.D. in applied mathematics and his M.S. from Iowa State University in 1967 and 1964, respectively. He received his B.A. from St. Mary's in Winona, Minnesota, in 1962. Dr. Zill also is former chair of the Mathematics Department at Loyola Marymount University. He is the author or coauthor of 13 mathematics texts.
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