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)

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)