# Problem sets and quizzes

About every other week, there will be a quiz, based on an associated problem set. Unless otherwise specified, all problems are from the 5th Edition of Differential Equations and Boundary Value Problems by Edwards et al published by Prentice-Hall (Pearson).

Here are the quizzes and their associated problem sets (Quiz 1, Quiz 2, Quiz 3, Quiz 4, Quiz 5):

1. Introduction:
• Date taken: July 23 Monday.
• Problems from §1.1 (pages 8&9): 3, 9, 16, 22, 25, 30, 36.
• Problems from §1.2 (pages 15–17): 3, 6, 10, 14, 22, 25, 30.
• Problems from §1.3 (pages 24–27): 2, 8, 12, 15, 21, 27.
• Problems from §1.4 (pages 40–44): 5, 10, 16, 17, 24, 31, 35, 39, 49, 61.
• Problems from §1.5 (pages 53–55): 3, 9, 16, 22, 30, 33, 37.
• Problems from §1.6 (pages 69–71): 7, 14, 29, 37, 47.
• Extra-credit essay question: Explain your background in mathematics and what you are going to use this course for. (Or just tell me if anything has changed since last term.)
2. Linear differential equations:
• Date taken: August 3 Friday.
• Problems from §3.1 (pages 147&148, note that Exercises 2–6 have accidentally been printed in the right-hand column): 3, 6, 9, 16, 18, 20, 22, 27, 33, 37, 40.
• Problems from §3.2 (pages 159&160): 3, 5, 8, 12, 14, 18, 21, 24, 26, 39.
• Problems from §3.3 (pages 170–172): 3, 7, 18, 19, 23, 26, 34, 37, 40, 43, 44.
• Problems from §3.5 (pages 195&196): 3, 4, 5, 10, 13, 22, 28, 31, 40, 43.
• Extra credit:
• Simpler problem: Exercise 3.5.54: Find the general solution of the differential equation g″(x) + g(x) = csc2x for 0 < x < π, using variation of parameters. (Either show what equations you solve to find the parameters or show what integrals you take to use a formula. Or show lots of work; you'll probably want to do that anyway.)
• Original problem: Following the development of Theorem 3.5.1 on page 194 of the textbook, find a general formula for y = g(x), given g‴(x) + P(x) g″(x) + Q(x) g′(x) + R(x) g(x) = f(x), assuming that P, Q, R, and f are continuous. Assume that you are given independent solutions y1 = g1(x), y2 = g2(x), and y3 = g3(x) for the corresponding homogeneous linear differential equation (in which f(x) has been replaced by 0); your answer will include some indefinite integrals that cannot be evaluated (much as Theorem 3.5.1 does).
3. Systems of differential equations:
• Date taken: August 15 Wednesday.
• Problems from §4.1 (pages 235–237): 3, 5, 11, 12, 14, 19, 21.
• Problems from §5.1 (pages 279–281): 2, 4, 6, 12, 18, 21, 24, 26.
• Problems from §5.2 (pages 293&294): 2, 5, 10, 29, 38.
• Problems from §5.5 (pages 346–348): 2, 4, 6, 23, 27, 30.
• Extra credit: Consider this system of differential equations and initial values:
• f′(t) = 5 f(t) − 4 g(t),
• g′(t) = 2 f(t) − g(t),
• f(0) = 3,
• g(0) = −1.
Calculate the exponential of the coefficient matrix of this system and use it to solve the system. (Hint: Look at Section 5.6.)
4. Numerical methods and applications:
• Date taken: August 27 Monday.
• Problems from §2.1 (pages 82–84): 2, 7, 9, 18, 21, 24.
• Problems from §2.2 (pages 91–93): 6, 10, 20, 21.
• Problems from §2.3 (pages 100–102): 1, 2, 4, 13, 14, 20.
• Problems from §2.4 (pages 113&114): 5, 8, 12, 14, 16, 19, 22, 23, 30.
• Problems from §6.1 (pages 380&381): 1–8, 13, 15, 16, 19, 20, 23.
• Extra credit: Use Euler's method with a step size of 1 to approximate f(3), where f is the solution to the differential equation f′(x) = f(x) with f(0) = 1. Then use the improved Euler method described in Section 2.5 of the textbook to approximate the same value. Which is closer to the actual value, e3 ≈ 20.1? (Show at least the numerical results at each step.)
5. Laplace transforms:
• Date taken: September 12 Wednesday.
• Problems from §7.1 (pages 445&446): 1, 8, 19, 21, 29, 32.
• Problems from §7.2 (pages 456&457): 1, 2, 3, 5, 6, 9, 11, 13, 16, 19.
• Problems from §7.3 (pages 464&465): 1, 3, 4, 6, 7, 8, 9, 11, 12, 15, 19, 27.
• Problems from §7.4 (pages 473&474): 1, 3, 5, 7, 9, 15, 17, 19, 22, 23, 26, 29.
• Problems from §7.5 (pages 482&483): 1, 2, 3, 5, 6, 9, 11, 13, 16, 19.
• Extra credit: Refer to Exercise 7.1.36 on page 446. Is the Laplace transform of f(t) = sin(exp(t2)) defined anywhere? (Explain why or why not.) Is the Laplace transform of f′ defined anywhere? (Explain why or why not.)
That's it!
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