### Homework assignments, Differential Equations (Math 256) Fall 2013

#### Prof. Janeba

NOTICE: There is a yellow bar somewhere below, which should cut this calendar in two somewhere around the current date.  Above the yellow bar are actual assignments.  Below the yellow bar are tentative assignments.  While the tentative items will mostly come true, there will be some changes.

Homework assignments for the week of:
Aug. 28 | Sept. 4 | Sept. 9 | Sept. 16 | Sept 23 | Sept 30 | Oct 7 | Oct 14 | Oct 21 | Oct 28 | Nov 4 | Nov 11 | Nov 18 | Nov 25 | Dec 2
Final Exam

### Week #1:

• Aug. 28 (This is what you should do after the class meeting on Aug. 28 and before the Aug. 30 meeting):
• Read text, sections 1.1 & 1.2, and the course syllabus
• Pay particular attention to the direction fields or slope fields in exaples 2 and 3 of 1.1.
• Also think hard about the method of example 2 in section 1.2.
• Prepare for a syllabus quiz on Friday.
• Do 1.1 #1, 11,15-20.
• For each of 1.1 #21,23,24:
• Identify the quantities that are changing in the problem, and what information the problem gives about their rates of change (derivatives).
• Use this information to write the requested differential equation, and complete the problem.
• Do 1.2 #3a, following the example 2 in the section.  Be careful, it's a bit different - so find a way to check your work.
• Finally, do the orientation for WebWork by shortly before midnight on Thursday.  You should have an email from me with step-by-step login directions.
• Aug. 30: Syllabus Quiz;
• Look at Prof. Janeba's survival guide to get important advice about reading a text.  This survival guide is written for calculus students, but it all applies to DE.
• Do 1.3 #1-5,7,9,11,12,13,15,19 (practice showing work for your quiz next Friday; hint for 15,19:  treat r as a constant, try to verify the specified solution, and note that this will only work for certain values of r.  What are those values?)
• Read 2.1 and do #13,15,19
• Start Integral Review 1 assignment on WebWork - due late next Wednesday, but you'll want to start now
• NOTE:  Exactly ONE of the problems in this WebWork is "too hard" - If you get all but that one, your grade will be considered 100% for this assignment (no matter what WebWork says)
The goal here is for you to learn to identify when the elementary methods of integration don't apply, rather than blindly using them despite the fact.

### Week #2:

• Sept. 2: Labor Day - no classes
• Sept. 4:
• Do 2.1 #14,17,20,21; Read 2.2 and do 2.2 #1,3,8
• There is a second WebWork integral review due late next Sunday night.
• Sept. 6: Quiz #1
• Do 2.2 #4,9,11,21, read 2.3 & ponder problem 2.3#1

### Week #3:

• Sept. 9:
• Read  "A word on writing solution verifications"
• Read  "A Math Course Survival Guide"
• Do 2.3 #1,3,4,13
• Do 2.3 #7,8 Hint:  Follow example 2.  Be sure to read what compounding continuously means; #7 can be done w/o differential equations, because the solution to the general problem is already given in the example, but #8 requires a new D.E. - follow the example to build it.  Note also that part 8c will require you to solve an equation computationally/approximately - an exact algebric solution is not possible.)
• Do the MOAMP (Mother of all mixture problems):
• Prof. Janeba's office initially contains 3000 gallons of Diet Pepsi (384,000 fluid ounces).  Diet Pepsi contains 35 mg of caffeine per 12 ounces.  Professors Johnson and Laison pour in coffee at a rate of 1200 fluid ounces per day; this coffee contains 100 mg per 8 ounces.  An unknown student lobs in (kind of like hand grenades) No-Doz at a rate of 30 pills per day; No-Doz contains 200 mg of caffeine per pill.
• Suppose, first, that all this fluid leaks out the window at the same rate as it is being added (in ounces per day).  You may assume the volume of the No-Doz is negligible, (though its caffeine content is not!)  i.e. the No-Doz does not contribute to the rate of fluid leakage.  You may also assume that the caffeine is "uniformly distributed" in the fluid in Prof. Janeba's office.  Then give an initial value problem for the amount of caffeine in Prof. Janeba's office as a function of time, AND give the amount of caffeine present in his office after 75 days.
• Extra Credit (optional, turn in neatly written by Friday's class if you desire) Modify the MOAMP into the GMOAMP: Assume that the fluid is leaking out at a rate of 3000 fluid ounces per day (inputs are unchanged from the MOAMP).  We still keep the "uniformly distributed" assumption. Then
• Give an initial value problem for the amount of caffeine in Prof. Janeba's office as a function of time (beware, the volume of fluid in the office is not constant), AND
• give the amount of caffeine present in his office after t days.
• For a super-bonus, use your answer to the previous part to tell what the concentration of caffeine is in the office just as the volume of fluid it contains reaches zero.  There is an intuitive answer here, but to get the super-bonus, you have to show how the previous answer shows it as well.
• Sept. 11:
• Do 2.3 #23  Note this is a very complicated problem, though very practical.  Solve it in stages.  Note the D.E. changes when the parachute opens, so you'll need to solve separately before-and-after.
• Read 2.4.  This will be rough sledding, but focus particularly on the examples - not only on the work done, but the conclusions drawn about the nature of DE solutions.
• Do 2.4 #1,3-5,8,10, 13,15
• Sept. 13: Quiz 2  postponed to Monday
• Read 2.5 (this one is fun; we can get lots of qualitative answers with relatively little work, for a certain kind of diff. eqn.)
• Do 2.5 #1,3,5, and 7 (7 is especially fun).

### Week #4:

• Sept. 16: Quiz #2;
• Some routine homework:  2.5 #8-12
• 2.5 #14 This problem requires some thought; be careful distinguishing between the graph of f(y) in the phase plane and the graph of y(t).
• 2.5 #15a,16ab
• So how can semistable equilibria occur in the real world?  Do 2.5 #21, an application to fish farming, AND explain in a sentence or two what each of the h values in (a),(d), and (e) mean in practical terms to the fish farmer.
• Sept. 18:
• Do 2.5 #22,23  (use Maple, Wolfram Alpha, or the like for the integral in 22b)  Note #23 is a somewhat complicated two-step problem
• Do 2.6 #1-4
• Sept. 20:
• Quiz #3
• Do 2.6 #5-7,10-14,18 (optional:  19,20: which show how integrating factors can help, but doesn't show how to find them.)
• Read 2.7  (The basics of Euler's method will be on next week's exam, you'll want to read up on those basics.
• Try 2.7 #1,2 parts a,d only.  Part (b) is easy to do with a spreadsheet or a graphing calculator, if you use it carefully.

### Week #5:

• Sept. 23: Do 2.7 #1,2,11,17  You may skip the parts about direction fields, but do use technology of some sort to compute the Euler estimates.
• Review exercises and study advice:  Work on review problems, p. 132-133:
• Just identify a possible solution method (e.g. linear (integrating factor ), separation, exact) for #1-24.
Note the authors have made it tricky in that you will often need to do some factoring or other rearrangement to "see" what method applies.
• For example, you will need to factor #13 before you'll get anywhere.
•  Now go back and solve several of #1-24, making sure to do several of each type (integrating factor, separation, exact).  Don't forget to include several that have initial conditions.
• You can keep going to #32 if you like, I have checked that #1-24 can all be solved with our methods, although the integrals may be hard.
• Be prepared to write a long paragraph to explain how/why each of these methods work:  linear/Integrating factor, separation of variables, exact d.e.s, as well as the setup of Euler's method.  We have discussed all of these in class, except for separation (the easiest), which is covered well on the top half of p.44.
• Don't forget to review modeling with D.E.'s (chapter 2.3)
• Refer to the survival guide for overall study advice - 70% doing problems, 30% organizing the material / deciding what you need to study.
• Sept. 25: Exam 1 (full hour)
• Sept. 27:
• Use error extrapolation as covered in class to refine your Euler estimates in any three of the homework problems from 9/23 above.  Note that our treatment assumes that one cuts the stepsize in half from one estimate to the next, so watch out on 11d, where h went from .025 to .01; similarly for the last estimate on #17.
• Read 3.1, and do 3.1 #1-5,9-12

### Week #6:

• Sept. 30:
• Do 3.1 #14,15,17 (and the 3.1 homework above that you haven't already done).
• Read 3.2.  There's a lot here, try to form an overview, and not get overwhelmed in the details.
• Oct. 2:
• Re-read (really) 3.2 in light of today's lecture, AND
• read 3.4 (yes, we are jumping around a bit)
• Do 3.2 #1,3,5,7-9,15,21,24
• Notes:  #5 and #24 are important examples,
• Be sure to read the directions on 7-9 carefully
• Do 3.4 #1,2,3,8,11,12,15
• Oct. 4:  Quiz #4; Read 3.3 & 3.5 and do 3.3 #1-3,7-9,11,13,17-19

### Week #7:

• Oct. 7:
• do 3.5 #1,2,3 (Hint for #3: Try yp = (At+B)e-t ),
• #4 (Hint for #4:  you need to get out the sum of a constant and a trig function; use something for each in your guess for yp.)
• #13
• Oct. 9:
• do 3.5 #5,7,9,15,16,18, and read 3.6
• Extra Credit:  Do 3.5 #30 with this change: y''+2y'+5y = 5 when 0t /2 [and keep y''+2y'+5y = 0 for t > /2
• write up an explanation of your solution that a fellow student could understand, and turn in by Tuesday afternoon for up to 8 points.
• Oct. 11: Quiz #5; Read 3.6 (again!) and do 3.6 #1,5  Hint for #5:  Use an integral table (online or in a book) to find sec(t)dt.

### Week #8:

• Oct. 14: Read A review of integration by parts, with a handy shortcut and...
• do 3.6 #1,5 if you haven't already, and 3.6 #7,10,13,15,16
• Note:  For #13,15,16, the text method only applies to d.e.'s in the form y''+p(xy'+q(x)y = g(x); the de's in these problems are not yet in this form.
• Oct. 16:  Read 3.7; do 3.7 #1-3, 5-8,13,17,18,24.
• Oct. 18:  Mid-semester day (no classes)

### Week #9:

• Oct. 21:
• Study for exam.  Spend some time on the various "how things work" explanations - can you produce them on a blank sheet of paper?
• Do the 3.7 homework above. Don't stress overmuch (stress some) if you have trouble sorting the coefficients k and from the physical description, but do be able to otherwise set up and solve the differential equation.  It won't hurt to try the one electrical problem by following the text example.  It would be a good idea to have some idea how the three categories of underdamped, overdamped, and critically damped are related to our three categories of characteristic equations (distinct real roots, repeated real roots, complex roots).
• A review assignment from last year - !skip the higher-order DEs!, and some solutions to it.  NOTE that we have learned some additional things, e.g. the "less-sophisticated argument," that is not listed on this handout, we have not covered so much on oscillators yet.
• Oct. 23: Exam II, Read 3.8, do  3.8 #5,7,9,11
• Oct. 24 (Thursday): Project 1 due, 4:00 p.m.
• Oct. 25:
• do 3.8 #17,18, which illustrate the difference between damped and undamped oscillators, as we approach the resonant frequency.  Note that oscillators with very low damping will behave approximately as undamped oscillators, in the short run at least, so the lack of "perfectly undamped" oscillators in nature doesn't make the undamped case irrelevant.
• Read 4.1 & 4.2.  Note that virtually all our methods from chapter 3 transfer immediately to higher-order linear d.e.s with constant coefficients, so these are essentially review sections.
• Read this review of solving higher-order polynomial equations, including polynomial long division.
• Do 4.2 #11,12,13

### Week #10:

• Oct. 28:
• Do 4.2 #14, 16,17,29,31,32 and  4.3 #4,5,1,2 (recommended in that order)
• Read 5.1 - which is titled Review of Power Series, but which may be new to you!
• Do 5.1 #1 and 19 to test your understanding.
• Oct. 30:
• Do 5.1 #1,3,4,6,7 - include in these both the radius of convergence and the interval of convergence, though you needn't worry about whether the endpoints are included.
• Do 5.1 #21,23,24.
Hint: (1-x) anxn =1 anxn  - x anxn = anxn - anxn+1
• Read 5.2.  Do 5.2 #2,4,5,6  (you may skip part (c) in each of these, for now)
• Nov. 1:
• Homework to be added later.

### Week #12:

• Nov. 11:  Read  6.2.  Do  6.1 #16,17,  and 26ac (use what we learned today about integration by parts)
• Do 6.2 # 1,3,5 (use the table on p.317 for all of these; complete the square for #5),
• Do 6.2 #11,14,16 (use your work for #5 in doing #16)
• Nov. 13:
• Do 6.3 #1,3,7,11,13  (should be short; use table for 13)
• Do 6.3 #20, 23  (a little harder)
• Do 6.4 #1,3,5,9  (the real deal; use Laplace transform tables rather that computing directly)
• Nov. 15: Quiz #7
• Read 6.5, do 6.5 #1,2,3,6, and #13 is an interesting challenge.

### Week #13:

• Nov. 18:
• In addition to the many complex calculations we've learned for this section of the course, here are some more theoretical / less calculational things to study for the exam:
• Be prepared, on the exam, to explain:
• Why did we study series solution methods?  What new problems did these methods allow us to solve, or what old problems did they allow us to solve more advantageously?
• Same question for Laplace transform methods.
• Why did we sometimes need to use partial fraction decomposition in a Laplace transform solution?
• What conditions (very generous conditions that are easy to satisfy, but still conditions) are there on a function f(t) in order for f(t) to even have a Laplace transform?
• A series solution in the form y = an(x-xo)n, for which we have worked out specific values for the coefficients an, will be valid on what sort of domain, in general?
• Nov. 20:  Exam 3
• Nov. 22:

### Week #14:

• Do an analysis of these systems of differential equations: draw a phase plane with some direction-field arrows drawn in and several solution trajectories sketched in, as we have done in class.
1. dR/dt = -JdJ/dt = -R
2. dR/dt = RdJ/dt = J  (in this case, yes, we could solve these separately - and trivially - but I still want the phase plane and trajectories traced.)
• Nov. 27:
• We did this worksheet in class for extra credit; if you didn't already do it in class, you may turn it in no later than the beginning of class on Monday, Dec. 2.
• Do 8.1 #1ab,3ab,use Error extrapolation as discussed in class to get better answers for #1&3 using your initial answers, then do 25bc,26, and ...
• Given the differential equation in #3, i.e. y'=2y-3t, DO NOT SOLVE, but find a formula for y" in terms of y and t by differentiating the DE itself with respect to t, then being creative.
• Nov. 29: Thanksgiving Holiday, no classes

### Week #15:

• Dec. 2:  Do 8.2 #1ab,3ab (notice how these are the same DEs from last week's 8.1 #1,3, using the Heun method, respectively.  Compare your answers, Euler vs. Heun.
• Ideally, you'd use Excel or a programmable calculator to cut down the amount of repetitive calculations in the homework above.
• For 8.2#3, we have these (Heun) estimates for y(0.4):
 stepsize Heun estimate 0.05 1.9056972 0.025 1.9062066 0.0125 1.9063397 0.00625 1.9063737
• Since we know that, for sufficiently small step size, the error in Heun's method is approximately proportional to (stepsize)2, or h2, then halving the stepsize does approximately what to the error?
• Put another way, halving the stepsize removes what part of the error?
• Use this to estimate the amount of remaining error in each of the h=0.025, h=0.0125, AND h=0.00625 Heun estimates.
• Now give a better estimate of y(0.4) that uses the h=0.05 and h=0.025 estimates above, i.e. extrapolate these estimates to give a better one.
• Repeat for the h=0.025 and h=0.0125 estimates, then again for the h=0.0125 and h=0.00625 estimates.
• You should now have three extrapolated estimates for y(0.4).  Comparing them to each other and to the estimates given in the table above,
• Do you find your extrapolated estimates more accurate?  (They should be, if the step size is small enough, what evidence is there that they actually are?)
• Based on all the above, do you believe the step sizes in the table above "are small enough?"
• Review problem:  Solve the D.E. in 8.2#3 exactly using chapter 2 methods (there will be an integration by parts involved).  Find y(0.4) exactly, and compare with the various estimates above.
• Some answers for today's problem.
• On your final exam next week, you may bring one 8.5"×11" sheet of paper, handwritten on one side with whatever you wish to write.  Do not bring computer-printed sheets, do not bring photocopied sheets, the sheet must be handwritten.
• Spend, as always, 70% of your study time actually doing problems.  Spend the other 30% organizing your thoughts about the categories of problems we have solved (and other theoretical questions we have studied this term), and the tools we have learned to solve them (and the theory we've learned).  Part of that 30% will be writing your sheet of paper.  If you expend some serious thought about what should go on the sheet, and how to organize it, (rather than writing everything you can find in really tiny print), it will be study time well-spent.
• A Laplace transform table will be provided on the final exam, the same table you had for exam 3.
• The final exam from fall 2007 is available on the class handouts page.
• Dec. 4:
• Quiz 8!
•  THE YELLOW BAR - Above this line are actual assignments.  Below this line are tentative assignments that might change before they become actual assignments.
• Read 7.1 and do 7.1 #1,2,3,5,15
• Find the equilibrium solutions to each system of differential equations:
• a) dx/dt = 1.5x-0.5xy  and dy/dt = -0.5y+xy
• b) dx/dt = 2x-5y-1  and dy/dt = -x + 3y
• Find dy/dx for both of the two preceeding systems, in general and at (1,2).
• Here's a real classic problem:  When physicists want to model a simple pendulum, they quickly get this differential equation:
l"+g·sin() = 0, where l is the length of the pendulum, g is the acceleration due to gravity, is the angle the pendulum string makes with the vertical, and the derivatives are with respect to time.
• This is a nasty non-linear D.E. (why is it non-linear?), so physicists make the classic approximation that for small values of , sin(), which changes the DE into a really nice, simple one
• Make the approximation noted above, treat g and l as constants, and solve the resulting second-order linear DE.
• But wait!  What if is not small?  Then the approximation above doesn't apply.
So transform l"+g·sin() = 0 with intial conditions (0) = /2, '(0) = 0 into a system of two first-order DEs with appropriate initial conditions.
• Now, using stepsize h=0.1, pendulum length 1 (meter), and g = 9.80 (m/sec2) and remembering to set your calculator on radians for the sine function,
find (0.4) using Euler's or Heun's methods.  Compare with the small-angle result above.
• (Heun's method with this stepsize has about one sig-fig of accuracy, two if you halve the stepsize.  The small-angle result, not so much.) (Answer: The Euler result is 0.983266546, which is high by about 25%.)
• Dec. 6:
• On your final exam next week, you may bring one 8.5"×11" sheet of paper, handwritten on one side with whatever you wish to write.  Do not bring computer-printed sheets, do not bring photocopied sheets, the sheet must be handwritten.
• Spend, as always, 70% of your study time actually doing problems.  Spend the other 30% organizing your thoughts about the categories of problems we have solved (and other theoretical questions we have studied this term), and the tools we have learned to solve them (and the theory we've learned).  Part of that 30% will be writing your sheet of paper.  If you expend some serious thought about what should go on the sheet, and how to organize it, (rather than writing everything you can find in really tiny print), it will be study time well-spent.
• A Laplace transform table will be provided on the final exam, the same table you had for exam 3.
• The final exam from fall 2007 is available on the class handouts page.
• I will be on campus at least part of every day next week.  I'll try to post some hours.

### Week #16: Finals Week

• Saturday, Dec. 14:  Final exam, 8-11 a.m.