Catalog
Description:
Slope fields, firstorder 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  twosemester sequence in differential and
integral calculus
2. Multivariate Calculus  onesemester course in multivariable calculus
3. Elements of Matrix Algebra from a multivariable calculus
course
Topics covered in this course include:
1. The firstorder 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 firstorder differential equations: Euler,
improved Euler and RungeKutta 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 firstorder 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 firstorder 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 15 and selected sections of Chapters 67 and 910 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 problemsolving skills.


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  Accelerationvelocity 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