# Math 3740      Differential Equations and Linear Algebra

Text: Differential Equations and Linear Algebra 2nd edition by C. H. Edwards and D. E. Penney

Catalog Description: Slope fields, first-order differential equations and applications, linear differential equations, numerical methods, solutions of systems of linear algebraic equations, eigenvalues and eigenvectors, system of differential equations, and series solutions. The computer algebra system Maple will be used to explore some of these topics.

Prerequisites: A passing grade (C or above) in Math 2720. This means that we assume students in this course have mastered:
1. Calculus I and II - two-semester sequence in differential and integral calculus
2. Multivariate Calculus - one-semester course in multivariable calculus
3. Elements of Matrix Algebra from a multivariable calculus course

Topics covered in this course include:

1. The first-order differential equations, uniqueness and existence of solutions, symbolic methods of integration
2. Applications of differential equations for modeling of population dynamics and motion
3. Numerical methods for first-order differential equations: Euler, improved Euler and Runge-Kutta methods
4. Matrices, linear system s of algebraic equations and their solutions, vector spaces, bases and linear independence
5. Linear differential equations of higher order, general solutions, superposition principle
6. Homogeneous linear equations with constant coefficients, their solutions
7. Nonhomogeneous linear equations - methods of undetermined coefficients and variation of parameters
8. Applications of linear differential equations for modeling of mechanical vibrations and electric circuits
9. Linear first-order systems of differential equations, existence and uniqueness of solutions
10. The eigenvalue method for linear systems with constant coefficients
11. Matrix exponentials and linear systems, fundamental matrix solutions
12. Nonhomogeneous linear systems
13. Numerical methods for first-order nonlinear systems of differential equations
14. Equilibrium solutions of nonlinear systems. Stability and asymptotic stability of equilibrium solutions. Phase plane
15. Linearization of nonlinear systems near equilibrium. Mechanical applications
16. Laplace transform method for solving linear differential equations

These topics correspond to Chapters 1-5 and selected sections of Chapters 6-7 and 9-10 of the text.

Objectives:

1. Understand differential equations as an important tool for modeling of physical and engineering processes.
2. Understand symbolic and numerical methods for finding solutions of differential equations and for analysis of their behavior.
3. Understand structure of solutions of linear systems of differential equations and related concepts of linear algebra (linear algebraic systems of equations, eigenvalues and eigenvectors)
4. Understand Laplace transform methods for finding solutions of linear differential equations
5. Understand concepts of stability of equilibrium solutions of nonlinear systems and methods of linearization of nonlinear systems near equilibrium
6. Understand the possibilities of modern computer algebra systems Maple in analysis of differential equations and their solutions, numerical methods and visualization for solutions.
7. Improve problem-solving skills.

Schedule:   The following suggested schedule covers 32 sections in 44-45 unit of 50 minute classes (11 weeks). Leave about 2.5 weeks for expanding, testing and additional materials.

 Math 3740 Suggested Schedule Section Unit Contents and Remarks 1.1 1 Differemtial  equations and math models 1.2 1 General and particular solutions 1.3 1 Slope field and solution curve. This is a unit suitable for a Maple Lab 1.4 1.5 Separable equations and applications 1.5 1.5 Linear first order equations 1.6 1 Substitution and exact equations. Optional 2.1 1.5 Population models 2.2 1 Equilibrium solutions and stability 2.3 1 Acceleration-velocity models. Lightly: emphasize qualitative analysis 2.4 2 Eulers's method. Mention the ideas of the algorithms in sections 2.5 and 2.6 here. 3.1 1 Linear system 3.2 1 Matrix and Gaussian elimination 3.4 1.5 Matrix operations 3.5 1.5 Inverses of matrices 3.6 1 Determinants 4.1 1 R 3 4.2 1 R n and subspaces 4.3 1.5 Linear combination and independence. Students often have difficulty with independence. Explain it geometrically helps. 4.4 1.5 Bases and dimension 5.1 1 Second order linear equation 5.2 1 General solutions. Make sure students understand the principle of superposition. 5.3 1.5 Homogeneous eqautions with constant coefficients 5.4 1.5 Mecanical vibrations. 5.5 1.5 Nonhomogeneous eqautions. 6.1 1.5 Eigenvalues 7.1 1 First order systems. Emphasis how to reduce higher order equations to first order differential equation systems. 7.2 1.5 Matrices and linear systems. 7.3 1.5 The eigenvalue method 7.5 2 Second order system and mechanical systems 9.1 1 Stability and the phase plane. 9.2 1 Linear and almost linear systems 9.3 2 Ecological models. Maple Project: Your own wildlife conervation perserve 10.1 1.5 Laplace transform 10.2 1.5 Transformation of initial value problems

Approved by the Department Curriculum Committee 4/08