Math 30613 Differential Equations Homework

NOTE: Homework assignments will be posted bottom to top – the most recent assignment will be on top.

HW 24. Due Wednesday 12/7/22

I. Do 6.3 #15, 16, 17, 18, 28, 29, 31

II. Read Section 6.4 (we’ll cover this material on 12/5)

III. Do 6.4 # 2, 3

HW 24. Due Monday 12/5/22

I. Do 6.2 #8, 9, 11

II. Read Section 6.3

III. Due 6.3 # 11, 12

HW 23. Due Monday 11/28/22

I. Do 6.1 #7, 8, 9, 11, 13, 16, 19, 24

II. Read 6.2

III. Do 6.2 #1, 4, 5, 6

HW 22 Due Wednesday 11/16/22

I. Do 4.3#4, 10, 12, 18, 21, 24

II. Read 6.1

III. Do 6.1 #1, 2, 3, 6, 26

HW 21 due Monday 11/14/2022

I. Read 4.2

II. Do 4.2 #1, 7, 11, 12, 17 (hint: you might look at things like period and amplitude of the steady state solution and how quickly the graph gets to the steady state), 19

III. Read 4.3

IV. Do 4.3 #1, 9, 15, 16, 17

HW 20. Due Wed. 11/9/22

I. Finish reading 3.6

II. Do 3.6 #21, 23, 24, 36

III. Read 4.1

IV. Do 4.1 #2, 3, 6, 10, 14, 19, 31

HW 19. Due Monday 11/7/22

I. Do Section 3.4 #9, 10, 15

II. Read the stuff we skipped about harmonic oscillators (pages 156-161, all of Section 2.3, page 256-258, pages 275-277)

III. Do 2.2 #19ab (note – the system here won’t be linear – why not?)

IV. Start reading Section 3.6

V. Do 3.6 #1, 4, 7, 9, 10

HW 18. Due Wednesday 11/2/22

I. Simplify the following into the standard form for complex numbers, a+bi

a. (3+7i)+(2+5i)

b. (1-2i)-(2-7i)

c. (1+2i)(3-4i)

d. 1/(2-4i)

e. (2+3i)/(1-6i)

f. e^3-5i

g. e^(7+2i)t

II. Start reading Section 3.4

III. Do Section 3.4 #1, 2, 3, 4, 5

HW 17. Due 10/31/2022

I. Read 3.3

II. Do 3.3 #1, 2, 6, 9, 12, 23, 24

III. Why do the solution curves bend the way they do in Figure 3.16?

IV. The last section on page 293 says that sometimes we need to trust computation above common sense. In what way does the model wind up not conforming to common sense?

HW 16 Due Monday 10/24/22

I. Read Section 3.2

II. Do Section 3.2 #1abde, 2abde, 4abde, 7abde, 13, 14, 16

HW 15 Due Wednesday 10/19/22

I. Read Section 3.1

II. Do Section 3.1#1, 5, 6, 8, 24, 25, 28, 29, 34

III. Show that if A is a matrix and v,w are vectors then A(v+w)=Av+Aw (do this by writing out the expressions in terms of components)

IV. Show that if Y₁ and Y₂ are solutions to , then Y₁+Y₂ is a solution. This should be similar to the argument in class that if Y is a solution then so is kY. Make sure to explain why each equality is true.

V. Do Section 3.1 #35 – this is a very important problem. It shows that if two solutions Y₁(t) and Y₂(t) are linearly independent at time t=0 then they are linearly independent for all time. This means that if Y₁(t) and Y₂(t) can be used to form the general solution from which we can find the specific solution for any t=0 initial conditions then this same general solution can also be used to find the specific solution given initial conditions for any initial time t=t_o.

HW 14. Due Monday 10/17/2022

I. Read Section 2.4

II. Do 2.4#1, 2, 4, 6, 7, 9, 10, 13abc

III. In class we considered the lines ax+by=e, cx+dy=f. We assumed that b≠0 and d≠0 and showed under this assumption that the lines have the same slope if and only if

ad-bc=0. Now let’s look at the cases where b or d is 0. First, suppose b=0. We will then also assume a≠0 because otherwise ax+by=e is not the equation of a line. So, assuming b=0 and a≠0, show that ad-bc=0 if and only if the lines have the same slope. You may argue geometrically using what you know about slopes and lines. What if instead we assume d=0, c≠0?

IV. Solve the following systems of equations – give all solutions or show that no solution exists. For this problem, try to solve algebraically without using the general theory from class:

a. 3x+4y=10, x-2y=8

b. 3x+4y=10, 6x+8y=20

c. 3x+4y=10, 6x+8y=5

V. For each of the following algebraic systems, use the general theory from class to determine (without attempting to solve) whether or not it is guaranteed to have exactly one solution. If it is, state this and stop. If not, determine all pairs (e,f) for which there will be infinite solutions.

a. 4x+2y=e

3x-5y=f

b. 2x+4y=e

-x –2y =f

c. 5x+2y=e

15x+6y=f

d. 3x- y=e

y=f

VI. Rewrite each of the algebra systems of part V as equations about linear combinations of vectors.

HW 13. Due Wed 10/12/2022* (we’ll talk about this in class)

I. Read Section 2.2

II. For the following, when the book instructs you to use HPGSystemSolver, instead use the system tab from the web site https://homepages.bluffton.edu/~nesterd/apps/slopefields_beta.html

Do Section 2.2#2, 3, 5, 9, 11, 12, 13, 18, 21, 23, 25

III. Why would drawing direction fields be less helpful for understanding non-autonomous systems than autonomous ones? Hint: how would this change the metaphor of the parking lot from the book?

HW 12. Due Monday 10/10/2022

I. Read Section 2.1 up to page 156
II. Do Section 2.1 #1, 2, 3, 4, 7a, 8ab, 9, 10, 11, 13, 14, 15, 16a, 17

HW 11. Due Wednesday 10/5/2022

I. Read Section 1.9

II. Do Section 1.9 #4, 5, 8, 19, 24, 26, 27a - you should use integrating factors for all of these problems to get practice with that technique

HW 10. Due Wednesday 9/28/2022

I. Read Section 1.8

II. Do Section 1.8#1, 3, 5, 8, 10, 19, 20, 23, 30, 31

HW 9. Due Monday 9/26/2022

I. Read Section 1.6

II. Do Section 1.6 #1, 5, 8, 13, 17, 20, 22, 30, 31, 34, 39

III. Suppose dy/dt=f(y), y(t₀)=y₀ is an initial value problem with f(y) continuous. Suppose y₀ is greater than all equilibrium solutions and that the phase line says that solutions in this region must be increasing. Will the solution to this initial value problem necessarily exist for all time? If not, what else can happen?

IV. Given an example of an autonomous differential equation whose only equilibrium is a node.

V. Start reading Section 1.8

HW 8. Due Wednesday 9/21/2022

I. Read Section 1.5

II. Do 1.5 (I suggest doing the problems in the order listed) #11, 13, 14, 1 (this question and the next two are about where you can tell your solution might or might not go based on the other information given), 2, 3, 9ab, 18

III. Suppose the hypotheses of the existence theorem are satisfied for dy/dt=f(t,y) at the point (t_0,y_0). Then you are guaranteed a solution exists. For how long in time can you be sure it exists?

IV. If the hypotheses of the existence theorem fail at some point, can you be sure that a solution doesn't exist there?

V. Can two solutions to a differential equation ever cross? Explain why not or when this might be possible.

VI. Give an example of a situation in which you can be sure a solution to a differential equation will last forever.

HW 7. Due Monday 9/19/2022

I. Read Section 1.4

II. Do 1.4 #2, 3 (for 2 and 3 make the table yourself – do not use the App, though you can use a calculator), 6, 11

III. To avoid making an error in a numerical approximation to a solution to a differential equation, what would be a wise thing to do if your calculation gives you a large slope?

HW 6. Due Wednesday 9/14/2022

I. Do Section 1.1#18b

II. Do Section 1.2 #41, 42

III. Read Section 1.3

IV. For the following problems, wherever it says to use HPGSolver, use instead the web site at https://homepages.bluffton.edu/~nesterd/apps/slopefields.html. Do 1.3 #2, 3, 6, 7, 10, 13, 14, 16

HW 5. Due Monday 9/12/2022

I. Read Section 1.2

II. Do Section 1.2 #1, 2, 3, 5, 6, 12, 15, 28, 33, 39, 40

HW 4. Due Wednesday 9/7/22

I. Continue reading Section 1.1

II. Do Section 1.1 Exercises #1, 2, 3, 4, 14, 15, 17, 18 (Hint: think about the qualitative analysis of the logistic model we did in class), 21, 22, 23

HW 3. Due Wednesday 8/31/22

I. Continue reading Section 1.1 through page 9

II. Do Section 1.1 Exercises #6, 7, 8, 11

HW 2. Due Monday 8/29/22

I. The mass of the Earth is 5.972 × 10²⁴ kg and its radius is approximately 6,371,000 meters. The mass of Mars is 6.39 × 10²³ kg and its radius is approximately 3,390,000 meters. Suppose you are standing on Mars and you have a rock of mass 983 g. At time t=0 you throw the rock upward from an initial height of 2 meters (above the surface) and with an initial velocity of 3 meters/second (upward). Using the formulas, ideas, and work from class, answer the following questions.

a) Find the acceleration due to gravity near the surface of Mars (don’t look it up! derive it using the information above and the fact from class that acceleration due to gravity near the surface of the Earth is about -9.8 m/s²)

b) Set up a differential equation to tell you the behavior of the rock after you throw it.

c) Solve the differential equation to find a function that tells you the height of the rock as a function of time.

d) At what time does the rock hit the ground?

e) How fast is the rock going when it hits the ground?

f) Suppose instead that you throw the rock at time t=1. What is the answer to part (c) then?

II. Start reading Section 1.1 of the book

HW 1. Due Wednesday 8/24/22

I. Do this calculus review assignment for some calculus practice.