Class Meets MWF 9:30–10:20

Room Bilger Hall 335

See here for the syllabus.

## Announcements

• (12/6) The final exam will be Monday, December 10 from 9:45am to 11:45am. The final will be held in the usual classroom. See here for the full final schedule.

• (12/4) Here is the formula sheet you will have for the final. Remember that no notes are permitted!

• (11/27) Here are a bunch of practice questions from the textbook to prepare for the final exam:

• Checking solutions of ODEs: (p. 24) 2, 4; (p. 37) 1, 3, 4, 5

• Solving first-order ODEs: (p. 104) 15–50

• Solving linear higher-order ODEs: (p. 220) 1–35; (p. 231) 3–23, 28–33; (p. 240) 1–15; (p. 311) 12–21

• Series methods: (p. 546) 3–12

• Real world problems: (p. 112) 2, 3, 4, 8, 9, 19; (p. 124) 1–12; (p.130) 1–7; (p. 321) 1–18; (p. 343) 6–10; (p. 353) 7, 8; (p. 376) 1–12

• (11/7) The study guide for the second midterm can be found here. Answer key:

1. $y_3(x) = 3 - 3x + 3x^2 -8x^3/3 + 8x^4/4$

2. $y = \frac 32 \cos x - \frac 12 \sin x + \frac 12 e^x$

3. $y = c_1 e^{-x} + c_2 xe^{-x} + e^{3x}/16$

4. $y = c_1 e^{2x}\sin(4\sqrt 3 x) + c_2 e^{2x}\cos(4\sqrt 3 x)$

5. Use Euler’s formula $e^{iu} = \cos u + i \sin u$.

6. $(2 - e^{-2s} - e^{-4s})/s$

7. $e^{-x^2}/y^2 + 2\int e^{-x^2}\,\mathrm dx = C$

• (11/5) There will be no class next Monday (November 12), as the university is closed for (the day after) Armistice Day.

• (11/2) The second midterm will be Friday, November 16. The take-home portion of the second midterm will be due Wednesday, November 21. The final exam will be Monday, December 10 from 9:45am to 11:45am. The final will be held in the usual classroom. See here for the full final schedule.

• (10/24) As I mentioned at the beginning of class, I’m doing an informal mid-semester evaluation to give you an opportunity to give feedback on my teaching. You can find it here. Participation is optional, but please respond by next Monday. As a reminder, the Google form is set up to not record your email address. Mahalo!

• (10/24) Math majors: The group advising session is the coming Monday, from 5 to 6:30pm. See this flyer for more information.

• (10/8) The Intergovernmental Panel on Climate Change recently released a report which shows that there is a high risk of climate change-related crisis as early as 2040. For a summary of the report, see this New York Times article.

• (10/4) Visualizing complex functions is hard, since their inputs and outputs are both points in the plane, and we only have three spatial dimensions in our world. A clever workaround is to use colors. Hue represents the argument of the output while intensity represents the norm of the output. Check out this great webpage by Hans Lundmark at Linköping University. His page explains how this works, and gives visualizations of some common complex functions, including the exponential and trig functions. It also mixes in some nice explanations about some cool facts about complex analytic functions, supported by the visualizations.

• (9/19) I put up a study guide for the first midterm. (9/24: fixed a couple typos.) Solutions: (3) $\log(6x^2 + 3y^2 + 4x + 2y + 1) = \sqrt 2 \arctan(\sqrt 2 \frac{x+1/3}{y-1/3}) + C$. (4) $y^4/4 - e^y = x^3 + e^x + C$. (5) $y^4(4x/y + 1)^3 = 1$. (6) $x\sin y + y \cos x = 0$. (7) $ye^{x+y} + e^x = C$.

• (8/22) The university will be closed Thursday and Friday, so we will not meet Friday and Friday’s homework will be due the subsequent Monday.

• (8/20) Tentatively, the midterm dates will be:

• Midterm 1: Friday, September 28.

• Midterm 2: Wednesday, November 21.

## Homework

I will grade from among the bolded or starred problems.

• Week 1 (Due Monday Aug 27)

1. Exercise 1.3 (page 5) from the textbook.

2. *** A petri dish of bacteria has 1000 cells at time $t = 0$. At time $t = 10$ it has 6 billion cells. Write an equation describing the size of sample at time $t$, assuming that the rate of growth is proportional to the size.

3. The inventor of chess was invited before the king, an avid chess player. Excited with the many hours of thrill he had obtained from the game, the king offered the inventor a reward. She replied that she was but a humble woman, and only wanted grains of rice for her reward. She asked for one grain of rice for the first square of the chessboard, two grains for the second square, four grains for the third square, and so on, doubling each time for the sixty-four squares of the chessboard. Thinking that he was getting a good deal, the king readily agreed to her request. How stupid was the king? Express your answer in number of grains of rice. Write a formula which gives the general solution for an $n$ by $n$ chessboard. Think about Carl Sagan’s statement that “exponentials can’t go on forever, because they will gobble up everything” (Carl Sagan, Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium, p. 17).

4. *** Read Lesson 2 of the textbook and review the material therein. Write the sentence “I read Lesson 2 and understand the material from that lesson.” and sign it with your name.

• Week 2 (Due Friday, August 31)

1. Exercises 3.1, 3.2.(a–e), 3.4.(b).

2. Exercises 4.1, 4.2, 4.3.

• Week 3 (Due Friday, September 7)

1. Exercises from Lesson 6: 2, 5, 7, 10, 11, 12, 15, 17, 20, 21

2. *** You have a 15,000 gallon pool. You want it to have the salinity of seawater, 35 parts per thousand of salt per water, which comes out to .3 lb of salt for each gallon of water. You miscalculated the size of your pool and put in 5500 lb of salt, which is too much. To fix this, you drain the pool at the rate of 5 gallons per minute while simultaneously filling it with fresh water at the rate of 5 gallons per minute. Meanwhile, you constantly stir the water to keep it the salt distribution uniform. Write a differential equation which describes the change in the quantity $s$ of salt in the pool (in pounds) at time $t$ (in minutes). Solve this differential equation to determine how long you need to drain the pool and refill it with fresh water to reach the optimal salt level of 4500 lb.

• Week 4 (Due Friday, September 14)

1. Exercises from Lesson 7: 1, 4, 6, 9, 10, 12, 14

2. Exercises from Lesson 8: 3, 6, 9

• Week 5 (Due Friday, September 21)

1. Exercises from Lesson 9: (Only have to solve by one method) 5, 7, 10, 16, 17

2. Exercises from Lesson 10: 3, 10, 12, 16

3. Exercises from Lesson 12: Do any five of 15–50

• Week 6: No homework due to exam.

• Week 7 (Due Friday, October 5): See this pdf. Canceled.

• Week 8 (Due Friday, October 12):

1. Exercises from Lesson 11: 2, 4, 10, 18, 21, 22.

2. Read Lesson 17M.E and do exercise 18 (page 189).

3. Exercises from Lesson 18: 6, 7. Think about, but don’t feel obligated to write down solutions for, 8 and 9.

• Week 9 (Due Friday, October 19):

1. Exercises from Lesson 20: 1, 3, 19, 31, 34.

2. Exercises from Lesson 21: 3, 13.

3. Exercises from Lesson 28A&B: 4, 18.

4. Exercises from Lesson 28C: 25.

5. Exercises from Lesson 29A: 10.

• Week 10 (Due Friday, October 26):

1. Exercises from Lesson 21: 5, 6, 7.

2. Exercises from Lesson 22: 10, 11, 14, 18.

3. Read Lesson 28D and do exercise 6.

• Week 11 (Due Friday, November 2):

1. Exercises from Lesson 27: 2, 4, 5, 6, 13, 14, 16, 19, 21.

• (Recall that $\sinh u = (e^u - e^{-u})/2$.)
• Week 12 (Due Friday, November 9):

1. Exercises from Lesson 37: 2 (hint: evaluate successive derivatives at $0$), 3, 6, 11. (You only have to use one method for each problem, but use the method of successive differentiation at least once and the method of undetermined coefficients at least once.)

2. *** Use the series method to find the exact solution to the differential equation $y’’ + y = 0$ satisfying $y(0) = 1$ and $y’(0) = 0$.

• You want calculate $y^{(n)}(0)$ for every $n$. Successively differentiating $y’’ + y = 0$ should reveal the pattern.

• Once you know $y^{(n)}(0)$ for every $n$ you then know the coefficients for the Maclaurin series for $y$. You should recognize the series.

• Hint: you can solve the ODE by previous methods to check your work.

• Week 13: No homework due to exam.

• Week 14: No homework due to exam/Thanksgiving.

• Week 15 (Due Friday Nov 30) See this pdf, which was the canceled week 7 homework.