BASIC KNOWLEDGE FOR MATH 221

SYMBOLIC TECHNIQUES

1. Solve separable first-order equations, obtaining explicit solutions where possible.



2. Solve homogeneous linear equations with constant coefficients.



3. Solve nonhomogeneous linear equations with constant coefficients using the method of undetermined coefficients.



4. Solve linear first-order equations by finding an integrating factor.



5. Write a second-order differential equation as a system.



6. 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.



7. Solve systems of two homogeneous linear equations with constant coefficients when the eigenvalues are real and distinct.

GRAPHICAL TECHNIQUES

1. Sketch minitangents in the slope field of a first-order equation or the direction field of an autonomous system of two first-order equations.



2. Sketch and interpret the phase line for a first-order equation.



3. Interpret the phase plane for an autonomous system of two first-order equations.



4. Use eigenvalues and straight-line solutions to sketch the phase plane for an autonomous system of two linear first-order equations.

CONCEPTS / MODELS

1. Discuss equilibrium points, stability, and interval of existence, and answer questions based on these concepts.



2. Identify whether or not a given method is appropriate for a given problem.



3. Solve word problems involving decay processes or unforced linear oscillators.