COURSE GOALS FOR MATH 221

Four goals that are common to all mathematics courses, including Math 221:

Math students should learn to

• think abstractly, analytically, and logically
• solve problems using knowledge, skills, intuition, and discipline
• recognize when a process is correct and complete
• communicate their work orally and in writing

Specific Goals for Math 221:

MODELING:

1. Learn to formulate differential equation models for physical processes and use them to explore questions of scientific interest.

CONCEPTS:

1. Understand the difference between a solution and a solution formula and know what conditions are necessary to guarantee that an initial value problem has a unique solution.

2. Understand the special properties of linear equations and be able to use the linearity principle to construct general solutions for linear equations.

3. Understand the relationships between solution curves, the slope field, and numerical approximations for first-order equations. Understand the notions of equilibrium solutions, stability, and the phase space for autonomous problems.

TECHNIQUES:

1. Learn to recognize and explicitly solve first-order equations that are separable or linear.

2. Learn to solve second-order equations with constant coefficients, using undetermined coefficients, variation of parameters, and the Laplace transform.

3. Learn to find equilibrium points and straight-line solutions for autonomous 2x2 systems and use a phase portrait to determine the stability of equilibrium points.

4. Understand the construction of the phase plane for two-dimensional autonomous systems, including equilibrium points, direction fields, and nullclines.

NON-CONTENT GOAL:

1. Acquire some experience using a computer algebra system.