# Ordinary Differential Equations Math 520 -- Fall 2013

## Homework Assignments

Date Assigned
Date Due
Section
Page
WileyPlus
Paper Prob
16 Jan 25 Jan §1.1 7 # 12, 13, 15, 17, 19 # 4, 5, 15-20
• For Problems 4 and 5, do not draw the direction field; instead, draw a direction line (as illustrated in class).
• Yes, do both 15, 17, and 19 on WileyPlus and 15-20 on paper. The practice will be good for you!
• Maple worksheet for direction fields
• Solutions
16 Jan 25 Jan §1.3 24 # 1-6, 15, 16 # 9, 12, 14, 28
23 Jan 28 Jan §1.2 16 # 2, 4a, 7ac # 4b, 7b
• For 4b and 7b you can give the answers for the specific equation that WileyPlus gave you for 4a and 7ac.
• See the Direction Field Plotter to help with comparing solutions in 1.
• Many of these questions introduce ideas that we will explore in much greater detail later in the course. Do not worry about trying to do more than you are asked to do - yet.
• Solutions
23 Jan 28 Jan §2.2 48 # 1, 4, 9a, 13a, 25 # 7, 10ac
• In #24, use the second derivative test to verify that the critical point is, in fact, a local maximum.
• The calculations in #31 should be simpler than the ones we did in class!
• Solutions
25 Jan 30 Jan §2.1 39 # 4c, 15, 20, 32 # 8(c), 16, 31
• Remember that the standard form for a first-order linear DE is $$y' + p(t) y = g(t)$$.
• Solutions
28 Jan 4 Sep §2.4 75 # 4, 5, 11, 14 # 25
1 Feb 4 Feb §2.6 99 # 3, 8, 10, 16 # 5, 6, 20
• Remember that the standard form is $$M(x,y)+N(x,y)y'=0$$ or $$M dx + N dy = 0$$.
• Solutions
8 Feb 8 Feb Exam 1 Chapters 1 and 2
(through § 2.6)
6 Feb 15 Feb §3.1 144 # 1, 2, 6, 10, 16, 23 # 7, 17, 24
• The basic form for solutions to linear homogeneous ODEs is $$y=e^{rt}$$
• $$\displaystyle \lim_{t\rightarrow\infty} e^{rt} = 0$$ when $$\Re(r)<0$$.
• Solutions
11 Feb 18 Feb §3.2 155 # 5, 6, 9, 18, 22, 24, 29 # 14, 28
13 Feb 20 Feb §3.4 172 # 2, 3, 8, ,14, 27 # 16, 18
15 Feb 22 Feb §3.3 164 # 9, 11, 19, 36 # 23, 31
• For #31, (13) is Euler's Formula.
• For #36, see the text provided with #34.
• Solutions
15 Feb 27 Feb §3.6 190 # 4, 10, 13, 31 # 11, 19
• For #11 and #19, because the RHS is not given explicitly, you won't be able to find an explicit formula for the general solution. Your answers will involve integrals whose integrands will involve the function g.
• Solutions
22 Feb 1 Mar §3.5 184 # 1, 2, 5, 11, 16, 20 # 12, 21, 25
• For #21b and #25b you should use Maple, WolframAlpha, or another tool that will give you the solution to the nonhomogeneous DE. Show this work (print a screenshot, etc.) and be sure you clearly indicate a particular solution of the given equation.
• Solutions
25 Feb 4 Mar §4.1 226 # 1, 4, 8, 10, 24 # 17
• In #1 and #4, if your answer involves more than one interval, enter your answer in the form $$(a,b) \cup (c,d)$$.
• Solutions
27 Feb 4 Mar §4.2 233 # 3, 11, 19, 31 # 8, 38
27 Feb 5 Mar §4.3 239 # 2, 12, 18 # 12
• Yes! I want you to work #12 twice. Once by hand and once in WileyPlus; they should be slightly different.
• Solutions
27 Feb 5 Mar §4.4 244 # 6, 13 # 4, 9
• The two hand-written problems are related; findin the general solution to the non-homogeneous problem (#4) is needed before you can find the particular solution satisfying a specific set of initial conditions (#9).
• Solutions
8 Mar 8 Mar Exam 2 Chapters 3 and 4
(except §§ 3.7 and 3.8)
• You may bring one notecard (not a full sheet of paper) on which you have written (i) the factorization of $$2r^4-r^3-9r^2+4r+4$$ and (ii) the solution of the linear system $$\left[\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 1 & 2 &-2 &-\frac{1}{2} \\ 1 & 4 & 4 & \frac{1}{4} \\ 1 & 8 &-8 &-\frac{1}{8} \end{array}\right] \left[\begin{array}{c} c_1 \\ c_2 \\ c_3 \\ c_4 \end{array}\right] = \left[\begin{array}{c} 2 \\ 0 \\ 2 \\ 0 \end{array}\right]$$.
• Exam 2
• Exam 2 Solution Key
18 Mar 25 Mar §7.1 361 #2, 9ab, 10a #14
• For #14, be sure to indicate what happens if $$a_{12}$$ and $$a_{21}$$ are both zero.
• Solutions
20 Mar 25 Mar §7.2 373 # 1a, 2c, 4bc, 10, 14. 21cd # 23, 26
• These should all be review of matrix manipulations.
• Solutions
20 Mar 25 Mar §7.3 385 # 3, 11, 13, 16, 24 (none)
• These should all be review from linear algebra.
25 Mar 3 Apr §7.4 394 # 6, 7 (none)
27 Mar 3 Apr §7.5 405 # 11, 15 # 24-27, 31
29 Mar 5 Apr §7.6 417 # 1a, 7, 16ab # 13, 14
• Phase Plane Analyzer is a new tool (in 2013) that I have started to create. It's written in Maple but you don't have to have a local copy of Maple to use it. It's still a rough draft. I will make changes as I have time. Please feel free to make suggestions.
• Solutions
1 Apr 8 Apr §7.8 436 # 1c, 7a, 9a # 1ab, 7b, 11a
• Worked solution to #8.
• For #1, use the specific problem you solved in WileyPlus to answer the questions for (a) and (b).
• For #7, use the specific problems you solved in WileyPlus to answer the question for (b).
• For #11, note the special structure of the coefficient matrix.
• Solutions
10 Apr 15 Apr §7.9 449 # 1, 3, 7 # 2, 6
• For #7 (WileyPlus) see if you can't make a wise guess at the form of the particular solution.
• Solutions
• Solution #3
17 Apr 17 Apr Exam 3 Chapter 7
19 Apr 24 Apr §9.1   # 1b, 4a, 10a, 11ab, 12a, 13, 15 # 4b, 10b, 12b
• Phase Plane Analyzer for Linear Systems is a new tool (in 2013) that I have started to create. It's written in Maple but you don't have to have a local copy of Maple to use it. It's still a rough draft. I will make changes as I have time. Please feel free to make suggestions.
• For the problems to be solved on paper, please use the corresponding problem presented to you by WileyPlus.
• In #1b, enter the eigenvector as a vertical vector with square brackets, e.g., $$\left[\begin{matrix} -1 \\ 2 \end{matrix}\right]$$.
• In #13 and #15, enter the critical point as a vertical vector with square brackets, e.g., $$\left[\begin{matrix} -1 \\ 2 \end{matrix}\right]$$.
• Solutions
22 Apr 24 Apr §9.3   # 5abc, 6a, 10a, 12a, 14abc # 6bc, 10bc, 12bc
• Phase Plane Analyzer for Locally Linear Systems is a new tool (in 2013) that I have started to create. It's written in Maple but you don't have to have a local copy of Maple to use it. It's still a rough draft. I will make changes as I have time. Please feel free to make suggestions.
• In #5a and #14a, enter the critical point as an ordered pair, e.g., $$( -1, 2 )$$.
• In #5b and #14b, all you have to enter is the $$2\times2$$ matrix, e.g., $$\left[\begin{matrix} -1 & 2 \\ 3 & -4 \end{matrix}\right]$$.
• WileyPlus does not have the correct answer in #10a. For all problems that I've seen, the correct answer should be $$(0,0)$$ and $$(-1,0)$$.
• In #12, recall that $$\sin(n\pi)=0$$ and $$\cos(n\pi)=(-1)^n$$ for any integer $$n$$. Consider two cases, even and odd, when analyzing the Jacobian.
• For the problems to be solved on paper, please use the corresponding problem presented to you by WileyPlus.
• Solutions
22 Apr 29 Apr §9.4   # 1bc, 3b, 13bcd # 1af, 3acf, 13a
• For #1a and #3a, sketch the nullclines and critical points instead of the full direction field.
• In #13bc, enter the critical points as ordered pairs, e.g., $$( -1, 2 )$$. Note that the answer in #13b will involve the parameter $$\alpha$$.
• For the problems to be solved on paper, please use the corresponding problem presented to you by WileyPlus.
• The solutions are being provided in advance of your work to give you an indication of the type of work I hope to see for the specific version of the problem that WileyPlus picks for you.
• Solutions
24 Apr 29 Apr §9.5   # 1f, 3b # 1abcde, 3acdef
• For the problems to be solved on paper, please use the corresponding problem presented to you by WileyPlus.
• The solutions are being provided in advance of your work to give you an indication of the type of work I hope to see for the specific version of the problem that WileyPlus picks for you.
• Solutions
6 May 6 May Final Exam Chapters 1, 2, 3, 4,
7, and (some of) 9
• The exam starts at 12:30pm and concludes 150 minutes later - at 3:00pm.
• Grades will be posted as soon as they are completed.

Notes:

• A useful tool for plotting the phase plane for a $$2\times2$$ homogeneous linear system is Java Applet for Phase Plane by Scott Herod at U. of Colorado.
• Maple worksheets (.mw files) should be downloaded to your local computer (I recommend creating a folder called, say, MapleFiles.)
• Portable Document Format (PDF) files are viewable with acroread, a publicly available PDF viewer by Adobe.
• PostScript (PS) files are viewable with ghostview, the public domain PS viewer.