The best way to prepare for a test
is to review homework problems, examples, definitions, and theorems from the
textbook and class notes. Make sure that you can solve problems in a reasonable
amount of time without reference to the textbook or class notes. It is
important that on the test you show all your work and explain your answers.
Just answers (especially wrong ones!) without any explanation will earn you no
credit.
List of major topics covered in
class.
1. Damped harmonic oscillator: equation of, guess and
test method, solutions in vector notation, solution curves (pp. 193-195). #15, 17 on p. 198.
2.
Linear systems: matrix notation (pp. 234 -240). Linearity
principle (pp. 243-246). Applications of Linearity principle to
solving initial value problems (pp. 246-249). Linear
independence and general solution (pp. 249-250). Undamped harmonic oscillator (pp.
250 -252). #5, 9, 11, 16, 17, 25, 27, 33 after 3.1.
3.
Section 3.2. Straight-line solutions, equation for finding
straight-line solutions, definition of eigenvalues
and associate eigenvectors, lines of eigenvectors. Computation of eigenvalues and eigenvectors, characteristic polynomial,
formulas for straight-line solutions. Linear independence of eigenvectors which correspond to distinct eigenvalues. #1, 2, 4, 7, 8, 11, 13, 16,
21, 23 after 3.2.
4. Section 3.3. Geometry of a
phase plane for linear systems with real eigenvalues:
saddles, sinks,
and sources. Stability of equilibrium point (0,0).
Given
a system,
you should be able to decide the type and stability of equilibrium point by
computing eigenvalues of the system. You should be
able to draw an approximate phase portrait of a system from the information
about its eigenvalues and eigenvectors (pp. 274-286,
class notes, handouts). # 1, 2, 4, 5, 8, 9, 11, 13, 14 after
3.3.
5. Section 3.4. Systems with
complex eigenvalues: Eulers's
formula, obtaining two real solutions from a complex solution. General solution in real and complex form. Theorems from
class: complex eigenvalues and eigenvectors of a real
matrix come in conjugate pairs, and real and imaginary parts of a complex
solution are also solutions.
Spiral sources, spiral sinks, centers. Natural
frequency, natural period, phase. Direction of spiraling: clockwise or
counterclockwise. Stability of equilibrium point. You
should be able to determine the type and stability of equilibrium point and
draw an approximate phase portrait by computing eigenvalues
of the system (section 3.4, class notes, handouts). # 1, 2,
3, 4, 5, 6, 8, 10, 11, 14, 17, 22, 23 after 3.4.
6. Additional review exercises: 1,
2, 4, 8, 11, 12, 13, 17, 19, 21, 27, 29, 31 on pp.
370-374.
GOOD LUCK!