**Math 415 , Autumn 2007, 8:30 & 10:30**

**
Instructor**:
Rodica D. Costin

**There is also (free!) tutoring offered at MSLC.**

The hand-outs for the 8:30 a.m. class and for the 10:30 a.m. class (the second page contains the assigned homework problems).

**W Sept.19**: topics: 1.1-1.3.

Homework: Read 1.3, and solve the homework problems listed on the hand-out:

p.8: 3,6,17; p.14: 1(a),(b), 3,6,11; p.22: 1,2,8,12,14,22,27

**F Sept.21**: topics: 2.1.

Homework: as listed on the hand-out: p.38: 14, 16, 18, 20, 28

**M Sept.24**: topics: 2.2 and from 2.3 only examples 1 and 4

Homework: less than listed on the hand-out:

p.45 **only** 4,6,7 and p.57 **only** 2,18,26

**W Sept.26**: topics: 2.4

Homework: less problem 32 (will be assigned next time):

p.72 **only** 4,9,14,24,27

**F Sept.28**: topics: 2.5 (autonomous equations, stable and unstable equilibrium points).

*Homework:* from 2.4 #32, and from 2.5 as listed on the hand-out: p.84: 1, 2, 7, 19, 23.

*Self-test *(7 minutes): find the general solution of the logistic equation and use this formula to show that

y=K is an asymptotically stable equilibrium point.

**M Oct. 1st**: topics: 2.6

*Homework:* as listed on the hand-out.

**W Oct. 3**: topics: 2.7, 2.8

*Homework:* as listed on the hand-out, **except** for problem 11(a), where just find y(t) first value of t, namely t=0.5.

**F Oct. 5**: topics: Review

*Homework:* as listed on the hand-out.

Special office hour: F Oct 5th, 12:30 to 1:10 in MW 436.

**M Oct. 8**: Test 1

*at the regular time and place as the lecture.*

**W Oct. 10**: planned topics: 3.1 and (most of) 3.5 - homogeneous equations with constant coefficients, the case of real characteristic roots

*Homework:* p.136: 1, 3, 5, 23, 33, 36, and p.166: 1, 3, 4, 9, 18, 23, 34

**F Oct. 12**: topics: 3.4

*Homework:* p.158: 3,4,5, 7, 11,12,15,29

**M Oct. 15**: topics: 3.4, continued.

It is helpful to know and use the fact that the general solution of a second order linear homogeneous equation with constant coefficients, whose characteristic roots

r=lambda+i mu

are complex (not real) can also be written in the form

A exp(lambda t) sin(mu t +B)

where A,B are arbitrary constants.

*Homework:* p.158: 17, 18, 19, 23, 39, 40

**W Oct. 17**: topics: 3.2

*Homework:* p. 159: 27 and p. 145: 3, 8, 18, 23, 27, 30

**F Oct. 19**: topics: 3.3

*Homework:* p. 152: 3, 4, 9, 15, 18, 20

**Please NOTE:** two solutions of a linear homogeneous differential equation y''+py'+qy=0

form a fundamental set (equivalently, are linearly independent) if and only if

W(t0) is not zero for some **t0 where p and q are continuous**, if and only if

W(t) is not zero for** all t where p and q are continuous**.

It may happen that W becomes zero at some points where p or q are not continuous (the equation itself has a problem at those points, there is no guarantee of existence or uniquness there).

**For the 8:30 class:** when calculating the Wronskian using W=C exp(-int p) we must first make sure the coefficient of y'' is 1.

Thus the Wronskian of the Bessel equation is C exp(- int(1/x)) = C/x.

**Hint for problem 30 p.146:** P(x)=x, f(x)=-1-cos(x).

**M Oct. 22**: topics: 3.6 Please **READ** Example 4 p. 173 and the paragraph above it. What it sais is that if you find

a particular solution y_p,1 of L[y]=g_1, and a particular solution y_p,2 of L[y]=g_2 then

y_p,1+y_p,2 is a particular solution of L[y]=g_1+g_2. (Note: L[y] is the same for all trhree equations!)

*Homework:* p. 178 as listed on the hand-out (1,3,4,5,6,9,13,17).

**Hints and comments on problem 23 p. 158** Find the formula for the solution. Plotting a (rough) sketch of its graph (oscillations between two exponential functions). Now be more precise: note that since u'(0)=0 then at t=0 the oscillation has either a min or a max. Find u''(0) and look at its sign to decide if it is a min or a max. Next, what is the "period" T of the ocillation ? Look on the graph and use your calculator to see between which values u increases or decreses. Note that u is zero at T/4 etc, decreases for t between 0 and T/2 etc. It is enough if you locate the first time when |u(t)|=10 within an interval of lenght T/4.

**W Oct. 24**: topics: 3.7

*Homework:* p. 183 1,3,7,13,18, and 29, 30 (one less problem than in the hand-out).

**F Oct. 26**: topics: 3.8, 3.9

Web resources: you can see the motion at the following sites

Simple Harmonic Motion

The Forced Harmonic Oscillator

*Homework:* p. 197: 2,6,17, and p.205: 1,5

(Note that there are two problems less than listed in the hand-out.)

**M Oct. 29**: topics: 5.1

*Homework* as listed on the hand-out.

Table of trigonometric identities

**W Oct. 31**: topics: 5.2 and Theorem 5.3.1 page 250, Example 3 p. 252.

*Homework*: In each of the following problems p. 247: 1,2,3,7,9

solve the equation by means of a power sries about the given point x_0 using the follwing steps:

1. Determine a lower bound for the radius of convergence of the series.

2. Find the recurrence relation for the coefficients.

3. Find the first four terms of the series and sort them to show the first four terms of two linearly independent solutions.

4. Find the first four terms of the series solution with y(x_0)=0, y'(x_0)=2 (for the given x_0).

**F Nov. 2**: topics: Review

**Special office hour:** Friday Nov 2, 12-1 p.m.

*Review* You should make sure you know how to solve problems of the following types:

3.1: 1,2,...,16, 22, 23,24,34,36,40-43

3.2: 1-6, 7-9,17,18,

3.3: 15-19,20,22,24,25

3.4: 1-23, 39-42

3.5: 1-14,20,23-30

3.6: 1-18

3.7: 1--11,10,12,13-20 (skip the ones with long calculations)

3.8: 1-4,5-7,11,13,14,17,19,20,24

Chapter 5: see the Homework assigned.

**M Nov. 5**: Test 2

**W Nov. 7**: 10.1

*Homework:* as listed on the hand-out.

**F Nov. 9**: 10.2

*Homework:* as listed on the hand-out.

**W Nov. 14**: 10.3 and Parseval's formula (see problem 17 p. 563)

*Homework:* p. 562: 1,3,6,8(a)

Web resource: Fourier series approximations

**F Nov. 16**: 10.4 and we started 10.5

*Homework: *p. 570 - 1,2,5,15,16 (one less problem than listed in the hand-out)

**M Nov. 19**: 10.5, 10.6

*Homework:* as listed in the hand-out.

**W Nov. 21**: 10.7

*Homework:* p. 600 1(a), 3(a) (only the first two problems listed in the hand-out)

Web resource: vibrating string Try plucking the string in different places, and see how the vibration changes. Slide the "damping" cursor to 0. Check the sound, too.

**M Nov. 26**: 10.8

*Homework:* 1a, 2, 3a (thus solving for all four cases), and 8a with f(x)=1.

**W Nov. 28**: Review Chapter 10.

**M Nov. 30**: Review

Some Review problems and their answers.

Also, do not miss to review *first order equations:* separable, linear non-homogeneous, exact.

Please check the Final Examination Schedule.

Note: CLASSES MEETING at 8:30 a.m. have the final on Mon Dec 3, 7:30 - 9:18 AM, and

CLASSES MEETING at 10:30 a.m. have the final on Wed Dec 5, 7:30 - 9:18 AM.

Web resource: equilibrium