Math 527 Outline of topics for final exam

Think of the final exams as 2/3 (exam1 + exam2 + exam3 + systems of equations). That's about right to fit in the allotted two hours. There will be around eight problems chosen from this outline of topics.

1st order equations
- separable
- exact
- 1st order linear
2nd and higher-order equations
- homogeneous constant coefficient ( $y=e^{\lambda t}$ is your friend)
- judicious guessing
- variation of parameters
- power series
Laplace transforms
- definitions, properties, and transforms & inverses of simple functions
- s-translation, t-translation
- Heaviside and Dirac delta functions
- convolution
Systems of equations
- matrices, vectors, $Ax=b$ problems, and determinants
- the eigenvalue problem, how it arises from the ODE $x' = Ax$
- how to solve systems of ODEs with
  - real eigenvalues, distinct
  - real eigenvalues, repeated
  - complex eigenvalues (solutions expressed in both complex and real-valued forms)
- phase portraits