REVIEW SHEET FOR THE FINAL

Final exam will cover the material we saw in class and studied in homework, sections 2.1-2.5, 2.7, 3.1-3.7, 4.1-4.4, 5.1, 5.2. (Review definitions and theorems covered in class, as well as those highlighted in the book.) Below is the list of most fundamental concepts that we studied as well as questions related to them.

This is NOT a substitution for reviewing your notes, homework and quizzes that you took, but rather a complementary hand-out to help you organize the material that we covered.

Definitions:

- order of a differential equation

- linear differential equation

- autonomous diff. equation

- homogeneous diff. equation

- equilibrium solutions

- characteristic equation

- Wronskian

- fundamental set of solutions (for 2^nd or 3^rd order linear equations)

- linear independence of functions (2 or 3 functions)

- determinants of matrices 2x2 and 3x3

Concepts:

- linear differential equation        

- integral curves

- direction fields

- asymptotically stable and unstable solutions

- phase line

Methods:

- Separating variables

- Integrating factor

- Bernoulli equation: substitution

- 2^nd order linear equations with constant coefficients : real roots, complex roots, repeated roots

- Euler’s method
- 3^rd and higher order linear equations with constant coefficients : real roots, complex roots, repeated roots

- Method of undetermined coefficients (for non-homogeneous linear equations)
- Method of variation of parameters (for non-homogeneous linear equations)

- Solutions as power series

Theorems:

- Existence and uniqueness theorems: 1^st order linear equations, 1^st order non-linear equations, 2^nd order linear equations, 3^rd order linear equations

- Principle of superposition (with proof)

- Abel’s formula (with proof)

- Theorem of 4 equivalent conditions on a fundamental set of solutions of 2^nd order linear equations

- 2 theorems on the general solution of a homogeneous differential equation of the 3^rd order (theorems 1 and 3 from 11/10/04)