MATH 231 SPRING 2011- COURSE LOG
W 1/12 syllabus/course policies
first-order equations: examples, implicit solutions, linear vs.
nonlinear
HW:
(1.2) 10, 11, 27, 28 (due 1/21)
Correction: (item 6. of course
policies in the syllabus): four
of the five grades will count towards the course grade
F 1/14 Constant-coefficient first order
equations/the existence-uniqueness theorem
Examples:
(1.2) 18, 28, 29, 31
HW (1.2)
17, 29 (b) (justify) (due 1/21)
M 1/17 Martin Luther King Jr. (Holiday, no
classes)
W 1/19 Linear first-order equations/solution
method, linear e/u theorem
Examples
(2.3) 10, 22 HW(2.3) 7,11,19 (due 1/21)
F 1/21 Separable equations (2.2) / the phase line
Examples: 14,
30, p.34 (d)
HW: 9,
13, 17 (2.2) ; (e),(h) (pp. 35/36) due 1/28
M 1/24 Exact equations (2.4) Examples: 10, 22
HW (2.4) 11, 15,
21, 23 (due 1/28)
W 1/26 integrating factors/ substitutions and
transformations (2.6)
Examples:
2,10,18 (2.6)
HW(2.6)
11, 25 (due 1/28)
F 1/28 mixing problems, population models (start)
Examples: (3.2)
2, 6, population growth with migration
HW (2.6) 17,19
(3.2) 3,5,13 (due 2/4)
M 1/31 radioactive decay, heating/cooling
Examples: (3.2)
24, 12 (3.3) 2, thermometer placed outside, 12
HW (3.2) 25,
(3.3) 7,11 (due 2/4)
W 2/2 1st-order problems in mechanics
Examples: 10,
25 HW (3.4) 1, 9, 13, 24 (for practice only)
F 2/4 Problem session (based on student questions)
Solution to problem 11,
section 3.3:
According to equation (9), p.111, with a thermostat the equation for
the temperature is:
T'=K[T_e-T]+K_U [T_D-T].
Here we're given K=1/2 and K+K_U=3 (see p.112, in boldface)
So K_U=5/2 and the DE is (with T_e=35 and T_D=16):
T'=57.5-3T,
with solution (given T(0)=35):
T(t)=19.2+(T(0)-19.2)exp(-3t)=19.2+15.8 exp(-3t)
Setting T(t)=27, we find: t=(1/3)ln (15.8/7.8)=0.24 h, or 14 minutes
M 2/7 1st. exam. Included: all sections presented
in lecture (see above). Problems on
the test will be very
close to either (i) a HW problem (ii) a example in the text, or
(iii) an example
discussed in class
BRING A VALID PHOTO ID (or you won't be
allowed to take the test)
Exam
1 (problems)
Exam
1(solutions)
W 2/9 2nd order linear DE, const. coeff. /discussion
of Exam 1
HW (4.2) 1,5, 15, 19 (due
2/11)
F 2/11 2nd order linear DE; constant coefficients
HW (4.3) 13,17,23,25 (due 2/18)
M 2/14 Non-homogeneous 2nd order DE: undetermined
coefficients, superposition
HW (4.4) 13, 17, 19 (4.5) 29
(due 2/18)
W 2/16 Variation of parameters
HW (4.6) 1,5 (due 2/18)
F 2/18 Variation of parameters (integral formula using a
special solution of L[y]=0) / equations with variable
coefficients (E/U theorem, domain of the solution,
Cauchy-Euler equations)
HW (4.7) 1,3 (find the interval of definition of
the solution), 11, 13, 21 (due 2/25)
M 2/21 Variable coefficients: variation of parameters, reduction
of order
Examples (4.7) 38, 40,
48 HW (4.7) 37, 41, 45 (due 2/25)
W 2/23 Free mechanical vibrations (4.9) Examples: 8, 10
HW (4.9) 3,5,18 (due
3/4)
F 2/25 Forced mechanical vibrations (4.10) Ex: 12, 14
HW (4.10) 9, 10 ,15
(due 3/4)
M 2/28 Solution of systems by substitution (inc. how to find the
number of consts in the general solution).
HW (5.2) 5, 7, 9, 23
(due 3/4)
W 3/2 coupled harmonic oscillators: normal frequencies,
normal modes
HW (5.6) 1, 2, 3 (for
practice)
F 3/4: review (problems based on student questions)
Monday 3/7: Exam 2. Material included: lectures from 2/9 to 3/2
(sections from chapters 4 and 5 in text)
The lectures also included topics found in the "group projects" E, F, G
in Chapter 4 (pages 256 to 259).
M 3/7
Exam
2
solutions
W 3/9 Laplace transforms: examples, first properties
HW (due 3/11) 9, 11, 13, 31 (section 7.3)
F 3/11 Inverse Laplace transform
M 3/14 to F 3/18: SPRING BREAK
M 3/21: Solution of IVPs via Laplace transforms
HW (due 3/25) (7.4) 9, 23, 41 (7.5) 1, 5, 35
W 3/23 Laplace transforms of discontinuous functions
HW (due 3/25) (7.6) 9, 15, 33
F 3/25 Laplace transforms of periodic functions
HW (due 4/1) (7.6) 27, 45
Examples: 41, 46
M 3/28 Convolution
HW (due 4/1) (7.7) 3, 9, 27
Examples: 2, 8, 28, 35
W 3/30 Solution of systems via Laplace transforms
Practice problems: (7.9) 1, 3, 5
F 4/1 Review session (Laplace transforms)
M 4/4 EXAM 3. Material: Laplace transforms (Chapter 7, sections covered
in class)
A table of transforms will be
provided on the test
Exam
3
W 4/6 Power series, analytic functions, radius of
convergence
Problems (8.2) 9, 11, 17, 21, 31 (due 4/8)
F 4/8 Power-series solutions of second-order DE
Problems (8.3) 3,5,15,17,25 (due 4/15)
M 4/11 Power-series solutions
Problems (8.4) 3,5,9,15,25 (due 4/15)
W 4/13 Linear algebra topics (see 9.3)
F 4/15 Diagonalizable systems of DE: general solution
Problems (9.5) 11, 13 (due 4/25)
M 4/18 Diagonalizable systems: fundamental matrix
Problems (9.5) 19, 21 (due 4/25): find the fundamental matrix that
equals the identity when t=0.
W 4/20: non-diagonalizable systems, complex eigenvalues (n=2) (9.6)
Problems (9.5) 36 (9.6) 1, 5, 7 (fundamental solution equal to the
identity when t=0)
F 4/22 Spring recess (no classes)
M 4/25, W 4/27: non-homogeneous systems, applications; reduction of 2nd
order equations to 1st order systems (9.7)
Practice problems (9.7) 1,3,13,21, 35
F 4/29: review (chapters 8 and 9)
FINAL EXAM: Monday, May 9--8AM to 10 AM. Material included: Chapters 8
and 9 of the text (material seen in class only)