• Instructor: David Ragozin
  E-mail: rag@math.washington.edu 
  Office: C337 Padelford
  Phone: 206-543-1148
  Study Session and Office Hours: 10:30-1:30 Thursday in CMU B022 or by appointment
  

Announcements
Homework
Comments
Links of Interest

• Syllabus
  
  
• Basic formulae for review (handed out during first class).
  
  
• Sample prereq. problems
  
  
• Proficiency test: Friday October 4 in class. See handout.
  Second try Tuesday Oct 8 at 8am Padelford C-36
  Third try Thursday Oct 10 at 8am Padelford C-36
  
  • Proficiency test scores


• Midterm 1 : Friday October 18 in class
  
  • Midterm 1 scores


• Midterm 2 : Wed November 20 in class
  
  • Midterm 2 scores
  Redo at home and hand in Wed Nov. 27.



• There exists a useful Java-Applet on Robert B. Israels web page. The applet can be used with any web-browser that supports Java (you may have to download a "Java-Plugin" to enable Java in your browser). The applet can draw direction fields for simple DEs, draw approximate solutions to initial value problems and also draw graphs of arbitrary functions --very useful to check if a given function is really a solution of a DE.
  
• Software for Math 307 (courtesy of John Sylvester): direction fields, numerical solutions to first order ODE's and other material. See comments below.

Date	Sections	Homework
Mon 9/30	Chapter 1	Read handout; Read Chapter 1 and 2.1; do 1.1 #11,13,18
Weds 10/2	2.1	study for proficiency test; 2.1 # 5, 13, 15, 20; Read 2.2
Fri. 10/4	2.1 + Proficiency test	2.2 #2, 4, 9, 11, 23; Read 2.3
Mon. 10/7	2.3	2.3 # 1, 10, 14, 18 plus circuit problem (see below); Read 2.5
Weds. 10/9	2.5 + Turn in HW on Ch 1 and 2.1-2.3.	2.5 #3, 7, 19, 24 (read 22,23); Read 2.6
Fri. 10/11	2.6	2.6 #1, 2, 7, 8; Read 2.7
Mon. 10/14	2.7 plus 2.4 & 2.8 (briefly)	2.7 #1a,b,d, 4a,b,d; 2.4 #8; 2.8 #9; Read 3.1
Wed 10/16	3.1+ Turn in HW on Sections 2.4-2.8	study for Midterm; 3.1 (due 10/24) p. 136: 1, 9, 11, 12, 19
Fri. 10/18	First Midterm
Mon. 10/21	3.1 and 3.4	3.4 #1, 5, 7, 11, 18; Read 3.2, 3.5, go over Midterm solutions.
Wed 10/23	3.2, 3.5; Turn in HW on Sections 3.1 and 3.4	3.2 #9, 11; 3.5 1, 6, 11, 17, 25, 29; Read 3.6, 3.7
Fri. 10/25	3.6	3.6 #1, 2, 3, 6, 14
Mon. 10/28	3.7	3.7 #1, 4, 5, 9, 10, 29; Read 3.8
Weds. 10/30	3.7; Turn in HW on Sections 3.5, 3.6, 3.7	3.7 #5 with initial conditions y(0)=1, y'(0)=2, #5 but with initial conditions y(p/4)=0 and y'(p/4)=0, #10 with initial conditions y(1)=1 and y'(1)=0; 3.8 #8, #12.
Fri 11/1	3.8 Electric Circuits	#18; Add to problem #8: Graph the solution, find the period, frequency, phase and amplitude. Add to problem #12: Is the charge overdamped or critically damped or underdamped? Graph the current. Add: redo #12 with 0.3 henries and answer the same questions. Also find the quasiperiod. How many henries would be needed to make it critically damped? Graph the charge in this case. What is the transient solution? (or is there a transient solution?)
Mon. 11/4	3.8 Electric Circuits	You've been assigned to build a simple electric circuit consisting of an inductor of 1 henry, one capacitor, and one resistor connected in series with no voltage source. The observed charge is supposed to be a damped oscillation which reaches its first local maximum of 5 coulombs after 1/2 second, and reaches its next local maximum of 3 coulombs at 2 seconds. If the solution is graphed, then it oscillates between two exponential curves. To approximate the solution, assume that the maximum occurs when the graph hits the exponential curve. Find (not necessarily in this order): the initial charge on the capacitor, the initial current, the resistance of the resistor, the capacitance of the capacitor, and equations for the charge and current as functions of time. What is the quasi-frequency, quasi-period, and the phase? How long will it take before you can be assured that the total charge will be less than .01 coulombs? Check your answer by using the 2nd order ODE solution plotter (see comment below) The maximum is actually slightly before the curve hits the exponential curve, but its close.
Weds. 11/6	3.8 Springs and Weights + Turn in HW assigned Weds-Mon.	3.8 #5, 9, 10, 17, 19; Read 3.9; On problems 9 and 10, it asks for the time t so that the function is small for all larger times. You don't need to find the smallest t, but find some time t so that the function is as small as indicated for all larger times. See comment on units below.
Fri. 11/8	3.9	3.9 #1 (also graph solution), 5, 7, 9, 10, 18
Mon. 11/11	HOLIDAY
Weds. 11/13	3.9 + Turn in HW assigned Weds & Fri.	3.9 # 6, 8, 11
Fri. 11/15	3.9	Review Problems below (not to be turned in)
Mon. 11/18	Review	My notes on the review problem solutions.
Weds. 11/20	Second Midterm
Fri. 11/22	6.1
Mon. 11/25	6.1 & 6.2	Due 12/4: Table page 304: #7-11; p. 298 #27;
Weds. 11/27	6.2 + turn in exam problem	Here's a copy of the midterm #2. Due 12/4: p. 307 #1, 2, 3, 6, 8, 10; p. 304 prove #15
Fri. 11/29	HOLIDAY
Mon. 12/2	6.2	p. 307 #11, 13, 14, 15, 16, 17
Weds. 12/4	6.3 + turn in HW on 6.1 & 6.2	p. 314 #1, 6, 7, 11, 13, 16, 17, 18
Fri. 12/6	6.4	p. 321 #1, 9, 10, 19
Mon. 12/9	Kepler's laws- NOT COVERED. Review began. Some notes on solution to 6.4, #10 from another professor. My lecture transparancies covering the same problem.
Weds. 12/10	Review. Final includes 3 Laplace transform problems, 1 problem similar to Midterm 1, problem 2 and/or 3, 1 problem similar to Midterm 2, problem 2 and/or 3. Review solutions to Laplace problems, to Midterm 1, #2,3 and to Midterm 2 review problems and Midterm 2, #2,3.



Circuit Problem Section 2.3: A decaying EMF given by E=200exp{-5t} is connnected in series with a 20 ohm resistor and a 0.01 farad capacitor. Assuming the charge Q=0 at time t=0,

a. Find the charge and the current at any time.
b. Show that the charge reaches a maximum. Calculate it and find when it is reached.


Extra problems section 2.3: If you are a Mariner fan, you might want to try problem #32, and if you are a snowboarder or skateboarder you might be interested in #33.


Review problems for the second midterm
#1: Consider the initial value problem

t^2 y" -t(t+2)y' +(t+2)y = t^3 exp(-t), with y(1)=0 and y'(1)=2. Answer the following questions:
a) How many solutions are there? (why)
b) Where are the solutions valid? (why)
c) Note that y(t) = t is a solution to the homogeneous equation (check it)
d) Find the solution to the initial value problem.
e) Check your answer by plugging it into the original equation given above (not some intermediate step).
f) What is the behavior of the solution for large values of t?

#2: Suppose a 2000 lb. car compresses a spring 2 inches. The free vibration of this spring is damped by a shock absorber which applies a force of 3000 lbs if the (vertical) speed is 30 mph. Several members of the JV squad of the local football team push up and down on the bumper with a force equal to 200cos(10t).

a) Describe the motion of the bumper by setting up the differential equation (an initial value problem).
b) Solve the initial value problem
c) Is the bumper underdamped, overdamped or critically damped?(why)
d) If there were no shock absorber, what force frequency would cause resonance?
e) The captain of the varsity football team walks up to the scene (either having done his differential equations homework or else with some experience in the matter) and tells them they can make the car bounce higher by pushing up and down at a different rate. How long (in seconds) should the period of the forcing term be to create the largest oscillations?
f) Does the car come off the ground (the spring will no longer be compressed at the maximum vertical displacement.) (Nb: if the oscillations are big enough, the car will come off the ground and they can then move the car when it is weightless)
You may solve the problem by carrying three significant digits instead of using rational or algebraic numbers.

#3: A resistor, capacitor and inductor are connected in series with an alternating voltage source. The resistor has resistance 180 ohms, the capacitor has capacitance 1/280 farads, the inductor has inductance 20 henries, and the applied voltage is given by E(t)=10sin(t). Assuming no charge on the capacitor, but an initial current of 1 ampere at t=0, when the voltage is first applied,

a) find the subsequent current.
b) What is the transient term? Which portion of the transient term is dominant? Graph the transient term.
c) What is the steady state term? Graph it.
d) Give initial conditions which would cause the beat phenomena if the resistor is removed.

My handwritten notes on the solutions to the midterm review problems.
************************************************************
FINAL EXAM:
Section D: Friday Dec. 13, 8:30-10:20 in our usual room.
Section K: Friday Dec. 13, 2:30-4:20 in our usual room.
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Comments

• This page is updated a frequently, so you should click on "reload" in your web browser to be sure you have the latest version.
• There is an error in the solution manual answer to problem 24b in Section 2.5. They have an extra "nu" in the first line and "nu" seems to be missing entirely from the next line.
• Hints on using the Software for second order equations:
  Note that p can be entered using the two letter word pi;
  the natural logarithm of t is log(t), not ln(t);
  exp(t) is e to the power t;
  you need to use the * sign for multiplication;
  you can also simplify using a hand-calculator first.
  If your solution and equation agree, you will see only one curve, otherwise you'll see two.
  
• Be careful that units are consistent. Some students are having trouble with problem #10 on Section 3.8 because they are mixing constants involving feet and inches without making the conversion. Write the units as if you were doing algebra with the words. For example to change 10 feet per second to inches per second, multiply by 12 inches/1 ft (this ratio is one since 12 inches=one ft) and cancel "ft".


Links of Interest

• Tacoma Narrows Bridge Disaster