**BASIC KNOWLEDGE FOR MATH 221**

**SYMBOLIC TECHNIQUES**

- Solve separable first-order equations, obtaining explicit solutions where possible.
- Solve homogeneous linear equations with constant coefficients.
- Solve nonhomogeneous linear equations with constant coefficients using the method of undetermined coefficients.
- Solve linear first-order equations by finding an integrating factor.
- Write a second-order differential equation as a system.
- Determine equilibrium points for a system of first-order equations. Determine the stability of the equilibrium point at the origin for a system of two linear first-order equations.
- Solve systems of two homogeneous linear equations with constant coefficients when the eigenvalues are real and distinct.

**GRAPHICAL TECHNIQUES**

- Sketch minitangents in the slope field of a first-order equation or the direction field of an autonomous system of two first-order equations.
- Sketch and interpret the phase line for a first-order equation.
- Interpret the phase plane for an autonomous system of two first-order equations.
- Use eigenvalues and straight-line solutions to sketch the phase plane for an autonomous system of two linear first-order equations.

**CONCEPTS / MODELS**

- Discuss equilibrium points, stability, and interval of existence, and answer questions based on these concepts.
- Identify whether or not a given method is appropriate for a given problem.
- Solve word problems involving decay processes or unforced linear oscillators.