Differential Equations
8:30 MW Daily Log

Math 272-01
Spring 2019

week 1, week 2, week 3, week 4, week 5, week 6,
week 7
, week 8, week 9, week 10, week 11, week 12,
week 13
, week 14, week 15, week 16

Items in bold were taken for a grade. The abbreviation ANN stands for announcement. The abbreviation MSL stands for MyStatLab. (The abbreviation MML stands for MyMathLab. It is referred to but technically the class uses MyStatLab.) Underlined items are clickable; click on it and then print out the handout.
Date
Activities
January 14

ANN: You will see four assignments in MML that are due this upcoming Monday. They are "MML Interface Homework" which will cover the MML interface including how to enter answers, "Chapter 1 Skills Check Homework" which, like every chapter, will start us on the chapter by reviewing some needed skills, "Section 1.1 Homework" will cover the material we cover today in class, and "Section 1.2 Homework" covering Wednesday's section. The syllabus will explain how we will usually have MML due the Monday after the week we finish the notes in class. ANN: We will not have class Wed., Jan. 23. I have to cancel class; sorry for the inconvenience. School is also cancelled on Monday, Jan. 21 for MLK Day. In class, we introduced ourselves to each other and talked over some bits from the syllabus and class policies. I also handed out a handout with information from the math tutoring center. I wanted to cover the notes for 1.1, so we cut the syllabus bit short and talked through some 1.1 material. I would expect you to do the homework in MML as soon as you can but it is due Monday, Jan. 21 as the announcement earlier said.

January 16
I handed out a list of Greek letters to help you interpret as you go through the semester. We started to discuss section 1.2. We got halfway through so the MML homework is postponed but you could start it if you wanted. In class, we defined some terminology and then worked through problems numbers 1b and 2b from the book's homework (pg 13) as examples. We started to discuss number 16, got as far as using implicit differentiation to find dy/dx, but had to stop. We will continue next time. Remember that we do not have class next week at all.
January 21
No school: Dr. King Day
January 23
[January 25 is the last day for a full refund.] No class: Cancelled due to teacher sick day
January 28
We finished our notes for 1.2. We introduced the idea of section 1.3 but got no further in the notes for it.
January 30
School was closed due to weather. We will amend the schedule and carry on next week.
February 4
ANN: In MML, you will see a Chapter 2 Skills Check Homework. This was set to be due Feb. 4 but will now be due Feb. 11. [I wrote this on the board but forgot to say anything so I will repeat this next time.] In class, we finished the class notes for 1.3. I handed out Direction Fields for Class Examples which we used in class as we completed the examples for 1.3. I handed out Direction Fields Worksheet to be done for Wednesday. It will be collected. We started the class notes for 1.4, getting through an introduction of the topic, Euler's method for graphing the solution to an initial value problem.
February 6
ANN: In MML, you will see a Chapter 2 Skills Check Homework due Feb. 11. The idea is that it provides several pre-requisite problems to get you ready for chapter 2. In class, I collected Direction Fields Worksheet and we nearly finished the class notes for 1.4. We did numbers 3, 7, and 10. Technically, we did not check our last answer to number 10 but we will next time. You should be able to start the MML 1.4 homework and even maybe do all of it. However, we will do an application next time to finish out the notes. I did hand out Euler’s Method for Approximating Function Values to be done for Monday.
February 11
ANN: I will try to teach more like a higher math instructor and less like a "math-education geared" instructor. That means I will not walk around the room or ask for input as much. But please do ask questions if they arise, in class and in office hours. ANN: In MML, you will have the Section 1.3 Homework and Chapter 2 Skills Check Homework due tonight, Feb. 11. In class, I collected Euler’s Method for Approximating Function Values. We discussed the issue that caused most people to get number 2c wrong from the worksheet that was due last time. The key was that the derivative (dx/dt) tells us if the solution function x(t) was increasing, and the second derivative has nothing to do with it. We finished up section 1.4 notes by checking our final answer to number 10 from last time. I decided to skip the last example for this section I had planned. We then went on to start chapter 2. I introduced the chapter by briefly talking through section 2.1. I suggest you read it if you want more of an introduction to the chapter. It is interesting. I handed out two documents that appear on the Assorted Handouts and Tutorials section of www.stlmath.com. There, at the bottom, I have links to some outside resources that will give you trigonometry and calculus resources such as cheat sheets with formulas and information. I handed out http://tutorial.math.lamar.edu/pdf/Common_Derivatives_Integrals.pdf and http://integral-table.com/downloads/single-page-integral-table.pdf but there are more to be gotten if you find you need them. I use them quite a lot as I do the problems. We then started section 2.2 notes. We defined a separable equation and wrote down the method by which we solve them. We then completed numbers 3, 1, 5, and 10 (in that order). We are working on number 14 but will need to finish next time. We used partial fraction decomposition to rewrite an integral so we could integrate in this problem. A review of partial fraction decomposition appears in Appendix A, page A-7 in the book.
February 13
We finished 2.2 notes by finishing up number 14. I reversed course on it a bit by not allowing ourselves to assume x > 0. We managed to solve the problem without that assumption. We had a good deal of discussion around knowing if a constant represented a real number or a positive real number or a non-zero real number. The differences do matter and you want to keep them in mind. We then went on to complete numbers 18 and 34. I mentioned that the homework might be a bit rough cause MML may have troubles in how you enter (correct) answers that are in a different form than they have gotten. You will need to tell me if that is a big issue. The MML problems for this section also do not include applications like number 34 we did today. I will probably supplement the homework with a worksheet after a few more sections of chapter 2.
February 18
We started the notes for 2.3. I introduced the section by defining a linear differential equation and then going over the method of solving them. I talked through some of the rationale but left most of it for reading. The book covers it on pages 49-50. As examples, we completed numbers 8 and 18. Before we did number 18, we wrote down the Existence and Uniqueness of Solution for initial value problems Theorem (pg 53) in our notes. We then started number 37 but really got only as far as writing down the problem and rewriting the differential equation in standard form, identifying P(x) and Q(x) which, using the variables in the problem, are to be written as P(t) and Q(t).
February 20
ANN: In addition to the MML homework, please complete the following on paper so that you can turn it in Monday, Feb. 25. Show work, circle your final answers, and please write down any integral formulas you used so I can follow. You are to do number 22 in section 2.2 and number 14 in section 2.3. In class, we finished the notes for 2.3 by wrapping up number 37 that we started last time. I made a comment about determining if an equation is linear by having to switch the roles of the variables to see if the equation is linear that way. This issue does not appear to come up until some problems in 2.4 which ask you to determine if an equation is linear. However, it is possible that you will see it in the MML homework for 2.3. We did not do an example because I doubt you will see the issue yet. We then started the notes for 2.4. I did mention that Khan Academy (www.khanacademy.org) is a great resource for diff eq information. To introduce section 2.4, we defined level curves and the total differential for a function. We defined the exact differential form and exact differential equations. We covered a test for determining if an equation is exact. We tried this procedure out for the equation given in number 7 in the homework (although we did not determine if it was separable or exact as directed in the instructions). We then started to write down the four step process for solving these differential equations. However, we only got through two steps so will continue next time.
February 25
Class cancelled due to teacher illness.
February 27
ANN: I gave out the Web address to Khan Academy wrong last time. It is www.khanacademy.org but is also linked from my home page under Links. ANN: Exam 1 will cover 1.1-1.4 and 2.1-2.6. You are allowed to bring a single page of notes. You can write anything you want on it but limit it to the front and back of a single sheet of paper. You should also bring your derivatives and integrals sheets I passed out. They are available on Assorted Handouts and Tutorials if you need to print out more. We will take the exam in the classroom on Wed., March 20. It will be paper and pencil. In class, we finished the notes for 2.4 by completing the method for solving exact equations that we started last time. We then did numbers 10, 26, and 29. Number 29 served as an introduction to section 2.5 which we then started. We started section 2.5 by writing the introduction of the section but did not get to any examples.
March 4
ANN: Exam 1 will cover 1.1-1.4 and 2.1-2.6. You are allowed to bring a single page of notes. You can write anything you want on it but limit it to the front and back of a single sheet of paper. You should also bring your derivatives and integrals sheets I passed out. They are available on Assorted Handouts and Tutorials if you need to print out more. We will take the exam in the classroom on Wed., March 20. It will be paper and pencil. In class, we nearly finished the notes for 2.5 by writing down an amended version of theorem 3 from page 68 and then doing numbers 2 and 8. I do have one last example (number 14) for our notes. However, instead of starting it, I gave out an assignment, due Wednesday. The assignment is to do numbers 6 and 10 from the book. Please, record any uncommon integral or derivative formulas used, circle final answers, and show your work. In addition, for number 6, write sentences to support conclusions and check for linearity with both variables as the dependent variable.
March 6
ANN: Mon., March 18 is a strict deadline for all work (paper or MML) collected in weeks 1-8. I will not accept late papers from section 2.5 or earlier after that date. In class, we finished the notes for 2.5 by doing number 14. I introduced section 2.6 by briefly saying we were going to see four types of equations that can be solved using substitutions or transformations. These will be turned into separable or linear equations which we know how to solve. We started out with our first type, homogeneous equations. We defined them and completed number 1 but only so far as to show it to be homogeneous. We wrote down the test for homogeneity and solution method described on page 71. We then started to complete number 10. We tested the equation to determine it is indeed homogeneous and then started to solve it, meaning we have just about shown it to be in the proper form dy/dx = G(v) where v = y/x. We will continue next time.
March 18
[March 11-15: No school: Spring Break] [March 18 is strict deadline for all late work from weeks 1-8.] ANN: Exam 1 is postponed to Monday, March 25. It will still cover chapters 1 and 2. As a result, you will see the homework will be due Sunday, March 24. In class, we finished number 10 for our example of a homogeneous equation. We then went on to cover how to solve equations in the form of G(ax + by), doing number 18 as the example for that. We, in fact, checked our solution as an added bonus. We then started the third type (of four covered in this section) of equation, Bernoulli equations. We wrote down the definition, comparing it to the linear equations seen back in section 2.3. We talked about how if n is 0, the equation is linear, and if n is 1, the equation is separable. We derived the method for solving a Bernoulli equation and then started to complete number 24. We got as far as dividing all by y^.5 but had to stop. We will continue next time.
March 20
ANN: Exam 1 will cover chapters 1 and 2. We will have it in class Monday, March 25. You are now allowed to use your whole notes as opposed to just being allowed one sheet on which to write notes. [I do not want you to reference your book as not everyone has one.] You should also remember to bring your derivatives/integrals sheets I handed out earlier. ANN: I assigned students to do, for Monday, numbers 6 and 20 from the book homework for section 2.6. In class, we worked through the rest of the notes for 2.6, completing number 24, writing up the fourth type of equation in the section which is equations with linear coefficients, and then trying to complete number 30 as an example of that. We did get a lot of number 30 done but will need to put the rest off until the Wed. after the exam. As a result, I will eliminate those problems from the homework and exam.
March 25
ANN: In MML, you will see Chapter 3 Skills Check Homework that is due tonight at midnight. In class, I collected numbers 6 and 20 from the book homework for section 2.6. We then took Exam 1.
March 27
We covered the last problem we had left behind for section 2.6, which was number 30. We then started the section 3.2. I mentioned that section 3.1 can be read as a nice introduction to the chapter, but it has no problems. For section 3.2, I introduced the idea of mixing problems (in a tank of salt water) with a picture and a generic differential equation. We then completed number 1 as an example. We then started the discussion of another topic in the section, population models. We looked at the differential equation that comes from the Malthusian (exponential) model for population growth. That led us to do number 9 even though it was essentially not a diff. eq. problem as we used the formula on page 96 (equation 11). It is actually just a problem as you might see in College Algebra since we did not need to do any integrating. But I wanted you to see it because it may have been a while since you would have done such a problem. We will continue section 3.2 next time.
April 1
We continued with 3.2 notes by completing number 19. We then talked briefly over the similarities between number 24 (or the like) which is a radioactivity example and number 9 which we did last week. You use the same base formula as we did in number 9 and it requires no differential equations per se. So, although we talked a little about number 24, we did not complete it. We went on to start the notes for section 3.3. I introduced the problem type and some base equations for the temperature inside a building and its rate of change. I did allude to Newton's Law of Cooling which you will see in early MML problems about coffee and beer. We then completed number 7 as our first example.
April 3
We continued with the 3.3 notes, doing number 8. We then started the notes for 3.4, getting through an introduction of the problem type with the important equations that we will use.
April 8
We finished the notes for 3.4, doing numbers 1 and 9 as examples. On number 1, we used the book's formula for x(t). However, on number 9 we could not and so needed to solve a differential equation. We then started the discussion of section 3.6 (skipping 3.5). We wrote out the introduction to the section, getting to the formula we will use for "y, sub n+1" from page 126. We will do the examples next time.
April 10
We continued section 3.6 notes by writing down the subroutine given on page 127 for the improved Euler's Method. We then completed number 8 by hand and number 10 by using the online calculator at www.math-cs.gordon.edu/~senning/desolver. We did explore number 8 some more by looking at its true solution (gotten by solving the separable equation) for these x-values. We then looked at the topic of Euler's Method with Tolerance. For an example, we used the online calculator to do number 12. I may have more examples for you next week. Steven mentioned that perhaps the online calculator I link to on the Website may not give as accurate answers as MML requires, so I will look into that. He mentioned the problem appeared when he did section 1.4 work.
April 15
ANN: I gave out the exams and asked students to make correction to them, turning them back in Wednesday. I will give half credit for deducted points. In class, we wrapped up the 3.6 notes by completing number 19 using the online calculator at www.math-cs.gordon.edu/~senning/desolver. We answered a further question, after writing out a table of the y values for every 6 hours for each of the k values given in the problem. We answered, "Which corresponds to the best insulation?" since we were supposed to be comparing insulation. We added to our table a column for the outside temperature, using the M(t) formula. We saw that k = .2 corresponded to the best insulation. We then went on to start chapter 4 by introducing the set-up in section 4.1. We started the first example, number 5. We are not done although we did find y' and y''. We will substitute those into the differential equation next time to show that y does in fact make it true.
April 17
ANN: Exam 2 will cover 3.1-3.4, 3.6, and 4.1-4.3 and is set for Monday, May 6, the Monday of the last week of class. In class, I collected the exam corrections. We wrapped up 4.1 notes by finishing number 5. We then defined a linear, second-order differential equation and then discussed synchronous solutions. We completed number 8 as an example of that. We then went on to start section 4.2 notes. I introduced the idea and talked at length about why the theorems of the section do really work. We completed number 2 as our first example. We then wrote down the theorem on page 161 about distinct real roots and foreshadowed theorem 2 (page 160) which we will get next time.
April 22
[Monday, April 22 is the last day to drop and receive a W instead of a D or F.] We finished the notes for 4.2, writing theorem 2 (page 160) and the definition of linearly independent (page 160) in our notes. We mentioned that y = 0 is always a solution (though not mentioned in the book's solutions) for every one of these equations in the form ay'' + by' + cy = 0. We also wrote down the repeated roots "theorem" on page 162. We then completed number 18 as our main example. We went on to discuss linear independence in more detail by doing numbers 31 and 29. We proved by contradiction the functions for number 29 are not linearly independent. I handed out Solving Homogeneous Linear Equations with Real Roots to be done for Wednesday.
April 24
I collected Solving Homogeneous Linear Equations with Real Roots. We then covered section 4.3 by going through a light version of the book's discussion on how they arrive at the main theorem which tells us how to find the general solution of the equation ay'' + by' + cy = 0 when the roots of the auxiliary equation are complex. We then completed numbers 6, 24, and 32. For number 24, we checked our solution.
April 29
ANN: All late paper or MML work is due by Mon., May 6. ANN: Exam 2 will cover 3.1-3.4, 3.6, and 4.1-4.3 and is set for Monday, May 6, the Monday of the last week of class. This will be the last day we meet. You will see that Chapter 3 and Chapter 4 Review Assignments in MML are due Sun., May 5. In class, we worked some review problems. Students had the opportunity to work the problems before we went over the solutions as a whole class. We worked on number 21 from 3.2 and started number 8 from section 3.4. We will finish this one next time, although I asked students to work on it before that to try to answer both questions.
May 1
ANN: All late paper or MML work is due by Mon., May 6. ANN: Exam 2 will cover 3.1-3.4, 3.6, and 4.1-4.3 and is set for Monday, May 6, the Monday of the last week of class. This will be the last day we meet. There will be about six questions on the exam. You will see that Chapter 3 and Chapter 4 Review Assignments in MML are due Sun., May 5. In class, we worked some review problems. Students had the opportunity to work the problems before we went over the solutions as a whole class. We worked number 4 from section 3.3. I mentioned that I would like to put something like it on the exam. We then finished number 8 which we started last time.
May 6
We took Exam 2.
May 8
Class reserved for testing in case of cancellation on May 7. See syllabus.