Math 345: Differential Equations | |
Course Description | Notes and Files |
Homework Assignments | Chapter Notes |
Course Description
The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modeling physical and other phenomena. Complementary mathematical approaches for their solution will be presented, including analytical methods, graphical analysis and numerical techniques. The basic content of the course includes
- first order equations
- mathematical models
- linear equations of second order
- the Laplace transform
- linear systems of arbitrary order and matrices
- nonlinear systems and phase plane analysis
- numerical methods
Course Contents
- Chapter 1 - Differential Equation Models.
- Chapters 2, 3, 6 - First-Order Equations and Applications: Solution techniques for linear, separable and exact equations. Modeling Examples. Stability of equilibrium solutions. Numerical methods.
- Chapter 7 - Linear Algebra and Linear Systems of Equations: Matrices. Matrix notation of linear systems of algebraic equations. Gaussian elimination. Subspaces and Bases. Determinants.
- Chapters 8, 9 - Systems of Differential Equations: General properties. Linear systems with constant coefficients. Eigenvalues, eigenvectors and characteristic equation. Fundamental set of solutions. Fundamental matrices and matrix exponential. Nonhomogeneous linear systems. The phase plane.
- Chapter 4 - Second-Order Linear Equations: Constant coefficient equations. Homogeneous and nonhomogeneous equations. Linear independence of solutions, characteristic equation. Superposition principle. Reduction of order, undetermined coefficients, variation of parameters. Applications from mechanical and electrical vibrations.
- Chapter 5 - The Laplace transform: Laplace transforms and their properties. Initial-value problems. Delta or impulse function and Heaviside or step function.
- Chapter 10 - Nonlinear systems: Linearization of a nonlinear system. Phase plane analysis and stability. Applications to biological model systems. Nonlinear Mechanics.