Differential equations with linear algebra

Abstract

1. First-Order Differential Equations. Differential Equations and Mathematical Model. Integrals as General and Particular Solutions. Direction Fields and Solution Curves. Separable Equations and Applications. Linear First-Order Equations. Substitution Methods and Exact Equations. 2. Mathematical Models and Numerical Methods. Population Models. Equilibrium Solutions and Stability. Acceleration-Velocity Models. Numerical Approximation: Euler's Method. A Closer Look at the Euler Method. The Runge-Kutta Method. 3. Linear Systems and Matrices. Introduction to Linear Systems. Matrices and Gaussian Elimination. Reduced Row-Echelon Matrices. Matrix Operations. Inverses of Matrices. Determinants. Linear Equations and Curve Fitting. 4. Vector Spaces. The Vector Space R 3. The Vector Space Rn and Subspaces. Linear Combinations and Independence of Vectors. Bases and Dimension for Vector Spaces. General Vector Spaces. 5. Linear Equations of Higher Order. Introduction: Second-Order Linear Equations. General Solutions of Linear Equations. Homogeneous Equations with Constant Coefficients. Mechanical Vibrations. Undetermined Coefficients and Variation of Parameters. Forced Oscillations and Resonance. 6. Eigenvalues and Eigenvectors. Introduction to Eigenvalues. Diagonalization of Matrices. Applications Involving Powers of Matrices. 7. Linear Systems of Differential Equations. First-Order Systems and Applications. Matrices and Linear Systems. The Eigenvalue Method for Linear Systems. Second-Order Systems and Mechanical Applications. Multiple Eigenvalue Solutions. Numerical Methods for Systems. 8. Matrix Exponential Methods. Matrix Exponentials and Linear Systems. Nonhomogeneous Linear Systems. Spectral Decomposition Methods. 9. Nonlinear Systems and Phenomena. Stability and the Phase Plane. Linear and Almost Linear Systems. Ecological Models: Predators and Competitors. Nonlinear Mechanical Systems. 10. Laplace Transform Methods. Laplace Transforms and Inverse Transforms. Transformation of Initial Value Problems. Translation and Partial Fractions. Derivatives, Integrals, and Products of Transforms. Periodic and Piecewise Continuous Forcing Functions. 11. Power Series Methods. Introduction and Review of Power Series. Power Series Solutions. Frobenius Series Solutions. Bessel's Equation. References for Further Study. Appendix A: The Existence and Uniqueness of Solutions. Appendix B: Theory of Determinants. Answers to Selected Problems. Index.