**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:

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

CONCEPTS:

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

- Learn to recognize and explicitly solve first-order equations that are separable or linear.
- Learn to solve second-order equations with constant coefficients, using undetermined coefficients, variation of parameters, and the Laplace transform.
- 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.
- Understand the construction of the phase plane for two-dimensional autonomous systems, including equilibrium points, direction fields, and nullclines.

NON-CONTENT GOAL:

- Acquire some experience using a computer algebra system.