Exercises

Exercise 9a - Due Friday, May 28

1. Section 10.1 # 9
2. Section 10.2 # 13, 14, 18, 28

Exercises 8b - Due Monday, May 24

1. Section 9.3 # 5, 6
2. Section 9.4 # 8, 9
3. Find the fixed points of the following system, classify them, sketch the neighbouring trajectories, and fill in the rest of the phase portrait.
   dx/dt = x-y, dy/dt = x^2 -4

Exercise 8a - Due Friday, May 21

1. Section 9.1 # 2ab, 5ab, 8ab
2. Section 9.2 # 5ac, 6ac, 9ac, 19

Exercise 7b - Due Monday, May 17

1. Section 7.5 # 7, 16
2. Section 7.6 # 2, 6, 13
3. Section 7.7 # [optional 7]

Exercise 7a - Due Friday, May 14

1. Section 7.1 # 21
2. Section 7.3 # 15, 16, 18
3. Section 7.5 # 1, 4, 5

Exercise 6b - Due Monday, May 10

1. Section 6.6 # 1, 6, 10, 12, 16, 18
2. Section 7.1 # 1, 3, 4, 10

Exercise 6a - Due Friday, May 7

1. Section 6.3 # 9, 10, 13, 16
2. Section 6.4 # 5, 9, 12
3. Section 6.5 # 3, 6, 12

Exercise 5b - Due Monday, May 3

1. Section 6.1, Exercises 6, 7, 26, 27.
2. Section 6.2, Exercise 16, 17, [optional 28], 30.

Exercise R1 - Review (will not be graded)

1. Section 2.5 Exercise 26
2. Section 2.10 Exercise 2,8,10,15,43
3. Section 3.5 Exercise 7,27
4. Section 3.6 Exercise 6,7,17
5. Section 5.2 Exercise 2

Exercise 4b - Due Monday, April 26

1. Section 3.7, Exercises 31, 32.
2. Section 5.2, Exercise 12.
3. Section 5.3, Exercises 6, 11, 18.

1. Section 3.6, Exercise 31, 32.
2. y₁ = x^-1/2 cos(x) is one solution of the Bessel equation
   x² y'' + x y' + (x² - 1/4) y = 0
   Use reduction of order to find the general solution.
3. Find the general solution of
   x² y'' + 2x(1-3x) y' + (9x² - 6x -2) y =0
   using reduction of order, given that y₁ = x e^3x is a solution.
4. Read Section 5.1 of Elementary D.E.s and B.V.Ps.
5. Section 5.2, Exercises 1, 7.

1. Section 3.4, Exercise 40.
2. Section 3.5, Exercises 6, 13, 41.
3. Section 3.6, Exercises 8, 9, 10, 15.

1. Section 3.1, Exercises 3, 9, 23.
2. Section 3.1, Exercises 31, 36, 40.
3. Section 3.4, Exercises 3, 6, 10, 14, 22.

1. Section 2.8, Exercises 18, 21.
2. Section 2.8, Exercises 26, 27.
3. Section 2.9, Exercises 4, 5, 15.
4. Section 2.9, Exercise 38.

1. Section 2.1, Exercises 13,14,28
2. Section 2.1, Exercises 29,31
3. Section 2.8, Exercises 3,4,8
4. (Hard) Show that the differential equation
   
   is not exact, and that the differential equation
   
   is exact. Explain why this is possible.

1. Section 2.4, Exercise 17. Section 2.5, Exercise 17.
2. Section 2.6, Exercises 1, 3, 10, 21.
   For the next two problems, refer to the graphs on the printed handout.
3. Describe the long term behavior of the solutions of y' = f(y), where f(y) is the function illustrated.
4. Find an equation whose solutions have the behavior illustrated.

1. Read Chapter 1 of Elementary D.E.s and B.V.Ps.
2. Section 2.3, Exercises 1, 3, 5, 9, 17.
3. Section 2.5, Exercises 6, 21, 27.