Fundamentals of Differential Equations

(Each chapter ends with a Chapter Summary, Review Problems and Group Projects.) 1. Introduction. Background. Solutions and Initial Value Problems. Direction Fields. The Phase Line. The Approximation Method of Euler. 2. First Order Differential Equations. Introduction: Motion of a Falling Body. Separable Equations. Linear Equations. Exact Equations. Special Integrating Factors. Substitutions and Transformations. 3. Mathematical Models and Numerical Methods. Mathematical Modeling. Compartmental Analysis. Heating and Cooling of Buildings. Newtonian Mechanics. Improved Eulers Method. Higher Order Numerical Methods: Taylor and Runge-Kutta. Professional Codes for Solving Initial Value Problems. 4. Linear Second Order Equations. Introduction: The Mass-Spring Oscillator. Linear Differential Operators. Fundamental Solutions of Homogeneous Equations. Reduction of Order. Homogeneous Linear Equations with Constant Coefficients. Auxiliary Equations with Complex Roots. Superposition and Nonhomogeneous Equations. Method of Undetermined Coefficients. Variation of Parameters. Qualitative Considerations for Variable-Coefficient and Nonlinear Equations. A Closer Look at Free Mechanical Vibrations. A Closer Look at Forced Mechanical Vibrations. 5. Introduction to Systems and Phase Plane Analysis. Interconnected Fluid Tanks. Introduction to the Phase Plane. Elimination Method for Systems. Coupled Spring-Mass Systems. Electric Circuits. Numerical Methods for Higher Order Equations and Systems. Dynamical Systems, Poincar? Maps, and Chaos. 6. Theory of Higher Order Linear Differential Equations. Basic Theory of Linear Differential Equations. Homogeneous Linear Equations with Constant Coefficients. Undetermined Coefficients and the Annihilator Method. Method of Variation of Parameters. 7. Laplace Transforms. Introduction: A Mixing Problem. Definition of the Laplace Transform. Properties of the Laplace Transform. Inverse Laplace Transform. Solving Initial Value Problems. Transforms of Discontinuous and Periodic Functions. Convolution. Impulses and the Dirac Delta Function. Solving Linear Systems with Laplace Transforms. 8. Series Solutions of Differential Equations. Introduction: The Taylor Polynomial Approximation. Power Series and Analytic Functions. Power Series Solutions to Linear Differential Equations. Equations with Analytic Coefficients. Cauchy-Euler (Equidimensional) Equations Revisited. Method of Frobenius. Finding a Second Linearly Independent Solution. Special Functions. 9. Matrix Methods for Linear Systems. Introduction. Review 1: Linear Algebraic Equations. Review 2: Matrices and Vectors. Linear Systems in Normal Form. Homogeneous Linear Systems with Constant Coefficients. Complex Eigenvalues. Nonhomogeneous Linear Systems. The Matrix Exponential Function. 10. Partial Differential Equations. Introduction: A Model for Heat Flow. Method of Separation of Variables. Fourier Series. Fourier Cosine and Sine Series. The Heat Equation. The Wave Equation. Laplaces Equation. (Chapters 11-13 are contained in Fundamentals of Differentials with Boundary Value Problems, Third Edition (0-201-33867-X).) 11. Eigenvalue Problems and Sturm-Liouville Equations. Introduction: Heat Flow in a Nonuniform Wire. Eigenvalues and Eigenfunctions. Regular Sturm-Liouville Boundary Value Problems. Nonhomogeneous Boundary Value Problems and the Fredholm Alternative. Solution by Eigenfunction Expansion. Greens Functions. Singular Sturm-Liouville Boundary Value Problems. Oscillation and Comparison Theory. 12. Stability of Autonomous Systems. Introduction: Competing Species. Linear Systems in the Plane. Almost Linear Systems. Energy Methods. Lyapunovs Direct Method. Limit Cycles and Period Solutions. Stability of Higher-Dimensional Systems. 13. Existence and Uniqueness Theory. Introduction: Successive Approximations. Picards Existence and Uniqueness Theorem. Existence of Solutions of Linear Equations. Continuous Dependence of Solutions. Appendices. Newton's Method. Simpson's Rule. Cramers Rule. Method of Least Squares. Runge-Kutta Precedure for n Equations. Answers to Odd-Numbered Problems. Index.