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Homework Assignments

Spring 2008

Directions: Please follow these directions on all homework assignments.

1. It is recommended, but not required, that you do all of your homework on engineering graph paper (available in the bookstore).
2. On each homework assignment, place your name in the top right corner of the page.
3. On the first line of the of the first page of your homework, please write down the assignment number, the pages that encompass the assignement, and list each exercise number assigned. For example, the first line of your homework might read: Assignment #12, Page 150, #1, 3, 5, 7, 8, 10, 11, 23, 45
4. If an assignment takes more than a page, please staple the pages together with a single staple in the upper left-hand corner.
5. Simple one or two word answers or choices without explanatory prose are not acceptable. In all cases, use sound writing to justify your response.
6. Please do not do computation for a problem on one sheet of paper, then refer to a graph or diagram on another sheet of paper near the end of your stapled packet. Keep your work together, compuations and graphs and diagrams in the same general neighborhood on your homework.
7. Please do not crowd your work on your paper. Space things out and avoid tiny diagrams that are hard to read (please be nice to my old eyes).
8. Assignments will be handed in during classtime in separate piles: the assignment #1 pile, the assignment #2 pile, etc., so please do not staple two or more assignments together.

• Assignment #1
• • Log onto Web Advisor and make sure that your personal information (email, etc.) is up-to-date. It is absolutely essential that your email address in Blackboard and WebAdvisor is current. Read the Syllabus, then reply to the Welcome Message in the Math 120 Discussion Board in Blackboard saying that you have completed all of these tasks.
• Assignment #2
• • Begin work on your Model Project. Due date for first draft is Feb. 8. Due date for second and final draft is Feb. 29.
  • Please seee the Model Page for a complete description of the Model Project, expectations, and due dates.
• Assignment #3
• • Read Chapter 1 in Differential Equations with Boundary Value Problems, page 1-14.
  • In Differential Equations with Boundary Value Problems, pags 14-15, do exercises #3, 6, 7, 10, 12, 13, 18, 20, 23, 25, 26, 27, 28. You may draw the graphs with any assistance you choose: Graphing calculator, Winplot, Matlab, or the Grapher application in the utilities folder on the Mac OS X.
• Video Resource
• • The video lectures at MIT Open Courseware are a great resource. You might try to view the first two lectures on this site. They are terrific!
• Student Term Project -- Timelines and Grading
• • The Student Term Project requirement is defined on the Syllabus.
  • You should carefully study due dates and grading structure on the Timeline and Grading page.
• Dfield
• • Install dfield on your system as follows:
    1. Create a directory called Polking on your hard drive.
    2. Download to your Polking directory the version of dfield at http://math.rice.edu/ dfield/ that matches the version of Matlab on your system. You can discover your Matlab version by entering the command ver at the Matlab prompt.
    3. At the Matlab prompt, enter the command pathtool. Add the directory Polking to your path. Save and close the pathtool.
    4. Test by entering the appropriate command at the Matlab prompt (e.g., dfield7 on Matlab 7 versions.
  • If you do not have Matlab on your system, you can use the Java version of dfield at http://math.rice.edu/ dfield/dfpp.html.
• Assignment #4
• • Read Chapter 3 of Ordinary Differential Equations Using Matlab, pages 28-32 only.
  • Read Chapter 2, Section 1 in Differential Equations with Boundary Value Problems.
  • In Differential Equations with Boundary Value Problems, perform the following tasks:
    • Solve exercise #17 on page 26 in two ways: (1) follow the directions used in the text, and (2) use the method of isoclines demonstrated in class and shown in the MIT Lecture 1 video on http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm.
    • Use dfield to draw the direction field for exercise #21 on page 26. Obtain a printout. Use a pencil to draw several integral curves on the direction field.
    • Use dfield to draw the solutions requested in exercises #27, 29, and 31 on page 26. Obtain printouts.
    • Use dfield to investigate exercises #37 and 38 on page 27. This may require the use of the zoom utility in dfield and you should probably submit a sequence of printouts that justify your answer.
• Assignment #5
• • Read Chapter 3 of Ordinary Differential Equations Using Matlab, pages 33-35 only.
  • Read Chapter 2, Section 7 in Differential Equations with Boundary Value Problems.
  • In Differential Equations with Boundary Value Problems, perform the following tasks:
    • Answer the questions posed in exercises #1 and 6 on page 86 and give reasons for your response. Use dfield to verify your answer. Obtain a printout. On your printout, explain what the image drawn by dfield implies in respect to the questions posed in each exercise which are also discussed near the end of the MIT Lecture 1 video on http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm.
    • Respond to the requests made in exercises #9 and 10 on page 86. Use dfield to demonstrate your response.
• Assignment #6
• • You will find Lecture #5 on http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm very helpful.
  • Read the remainder of Chapter 3 of Ordinary Differential Equations Using Matlab, pages 35-43.
  • Read Chapter 2, Section 9 in Differential Equations with Boundary Value Problems.
  • In Differential Equations with Boundary Value Problems, perform the following tasks:
    • Answer the questions posed in exercises #1, 2, and 4 on page 100, using dfield to draw the required sketches. Obtain a printout. Use the text capabilities to mark each equilbirum solutions with its equation.
    • On pages 100-101, draw the sketches requested in exercises #8, 10, 12, 14, 15, 18, 19, and 22 by hand.
    • On page 102, use hand calculations to supply the requests posed in exercises #27, 28, and 29.
    • Use hand drawn sketches to do the analysis requested in exercise #30 on page 102. Use dfield to verify your result. Include the equilibrium solution and several nonequilibrium solutions as well. Obtain a printout.
• Assignment #7
• • Read Chapter 2, Section 2 in Differential Equations with Boundary Value Problems.
  • Read Chapter 2 of Ordinary Differential Equations Using Matlab.
  • In Differential Equations with Boundary Value Problems, perform the following tasks:
    • Use dfield to draw a family of solutions for the differential equation posed in exercise #5 on page 35. Then use the separation of variables technique to find the general solution of the differential equation. Finally, use Matlab to sketch a family of solutions to the differential equation and compare with the result found by dfield. Obtain printouts of each.
    • Use dfield to draw the solutions to each initial value problems posed in exercises #13, 15, 17, and 21 on page 35. Then use the separation of variables technique to solve each initial value problem and state its interval of existence. Finally, use Matlab to sketch the solution found by the separation of variables technique and compare it to the solution found by dfield. Obtain printouts of each.
    • On page 36, do exercise #33.
    • On page 37, do exercises #37 and 40. Provide detailed hand drawn sketches for each exercise.
• Assignment #8
• • You will find lectures #3 and #4 at http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm. In lecture #4, only the part that addresses Bernoulli's equation will be useful for this assignment.
  • Read Chapter 2, Section 4 in Differential Equations with Boundary Value Problems.
  • In Differential Equations with Boundary Value Problems, perform the following tasks:
    • Do exercises #2 and 4 on page 55. Show detailed steps.
    • Use dfield to sketch the solution of the initial value problem posed in exercises #16 and 21 on page 55. Find analytical solutions, then use Matlab to sketch the plot of these solutions over the same domain you used in dfield. Obtain printouts of each.
    • Do exercises 22, 23, and 28 on page 55. Show detailed steps.
  • Read Chapter 2 of Ordinary Differential Equations Using Matlab.
  • In Ordinary Differential Equations Using Matlab, perform each of the following tasks.
    • On page 26, use dsolve to solve the initial value problem in exercise #16, then provide a pencil and paper solution and compare solutions. Use ezplot to sketch the solution on the given interval and obtain a printout.
    • On page 27, provide a pencil and paper solution of the given initial value problem in exercise #42 and show that your solution matches that given in the manual. Then provide the plots requested in the exercises and obtain a printout. Using prose, explain the terms transient solution and steady state solution in terms of what you see in your solution and plot.
• Assignment #9
• • Read Chapter 4 of Ordinary Differential Equations Using Matlab.
  • In Ordinary Differential Equations Using Matlab, perform each of the following tasks.
    • On page 58, perform the tasks outlined in exercise #17. Obtain a printout of the results.
    • On page 58, redo exercise #17. However, this time rewrite the script file as a function file that uses the given function definition as a subfunction. Follow the lead in the example given in the narrative on page 51 of Ordinary Differential Equations Using Matlab.
    • On page 59, use pencil and paper calculations to solve the given linear initial value problem in exercise #27. Verify that your solution matches the given solution in the manual, then perform the rest of the tasks requested in the exercise. Obtain printouts of each result as you vary omega. Use prose to explain the meaning of a low pass filter, connecting your description to the experiment outlined in this exercise.
• Assignment #10
• • Lecture #2 (Euler's Numerical Method) at http://ocw.mit.edu/OcwWeb/Mathematics/18-03Spring-2006/VideoLectures/index.htm is a helpful supplement.
  • In learning numerical solutions of ordinary differential equations, the main source we will use is Chapter 5 (Numerical Methods for ODE's) in the manual Ordinary Differential Equations Using Matlab. However, Chapter 6 of the text Differential Equations with Boundary Value Problems is a valuable supplement.
  • In Ordinary Differential Equations Using Matlab, perform each of the following tasks.
    • On pages 71-73, follow the directions in the manual exactly in exercises #1, 10, 16, 20, 21, and 22.
• Quiz #1
  • You can now download Quiz #1
• Assignment #11
  • In Ordinary Differential Equations Using Matlab, read Chapter 6. Note: It is important that you are seated at a computer, typing in the instructions in the chapter as you read along.
  • In Ordinary Differential Equations Using Matlab, perform each of the following tasks:
    • On pages 88-89, do exercises #8 and 9. Obtain printouts for each.
    • On page 89, do exercise #13. The exercise requires that you perform the experiment with input voltages having periods 32, 16, 8, and 4. Follow the directions precisely and obtain 4 separate plots, one for each input voltage. Keep the same domain and range for each plot, namely t in [0,96], V_c in [0,4]. Use multigraf.m to arrange the 4 separate plots on one page. Obtain a printout. Write a paragraph on your printout explaining the term "low pass filter."
    • On pages 90-91, do exercise #16.
    • On page 91, do exercise #17. Use multigraf.m to arrange plots for several values of H. In each case, explain what happens to the fish population.
    • On pages 91-92, do exercise #19. Use multigraf.m to arrange your results. Use writing to explain your conclusions.
• Assignment #12
  • In Ordinary Differential Equations Using Matlab, read Chapter 7. Note: It is important that you are seated at a computer, typing in the instructions in the chapter as you read along.
  • In Ordinary Differential Equations Using Matlab, perform each of the following tasks:
    • On pages 101-102, do exercise #13. Obtain separate plots for each value of the parameter H and arrange your results with multigraf.m. Write a paragraph explaining your conclusion.
    • On page 102, do exercise #14. Obtain a printout.
    • On pages 102-103, do exercises #23 and 26. Obtain printouts for each. Use print setup to position the plot at the top of the page, leaving space for your conclusions below your plot. In the space created, show how you changed the second order ode into a system of two first order odes.
    • On page 103, do exercise #27. Obtain a printout. You can find help in Differential Equations with Boundary Value Problems, Example 5.4, pages 60-61.
• Assignment #13
  • In Ordinary Differential Equations Using Matlab, read Chapter 8. Note: It is important that you are seated at a computer, typing in the instructions in the chapter as you read along.
  • In Ordinary Differential Equations Using Matlab, perform each of the following tasks:
    • On page 125, do exercise #4. Obtain a printout of your plot and the code that produced the plot.
    • On page 125, do exercise #8. Solve the equation analytically, then answer parts (a), (b), and (c). Obtain a printout of your plots and the code that produced your plots.
    • On page 125, do exercise #13. Do the problem dynamically as explained in the exercise, then collect the data and use the plot command to duplicate the dynamic results for printing. Obtain printouts of the plots and the code that produced your plots.
    • On page 126, do exercise #18. Arrange the plots for parts (a), (b), and (c) using multigraf. Obtain a printout of the plots and the code that produced them.
    • On page 126, do exercise #19b. Obtain a printout of the plot and the code used to produce the plot.
    • On page 126-127, do exercise #20. Arrange the plots for parts (a), (b), (c), and (d) using multigraf. Obtain a printout of the plots and the code that produced them.
• Interesting Links
  • Matlab's OutputSel key can be set using odeset to your own self built output function. This can be used to drive an animation. For an example of use, see the document: ODEs in Matlab, Worksheet #1

    Dr. Frank Schilder, a chemistry teacher, wrote this set of notes and exerrcises for his nonlinear dynamics class./p>

  • In Orbits Worth Betting On in the Winter 1996 issue of the Consortium for Ordinary Differential Equations Experiments (CODEE), you can get a sense of the difficulties inherent in Duffing's Oscillator, addressed in Example 5 on page 121 of Ordinary Differential Equations Using Matlab. You can examine other back issues of the CODEE newsletter at: http://ccdl.libraries.claremont.edu/col/cla/
• Quiz #2
  • You can download Quiz #2. The quiz is due Monday, March 10, at the beginning of class.
• Assignment #14
  • Download Odesolve, a program written by John Polking, from the Rice site. Place it in the same folder containing dfield7 and friends.
  • In Ordinary Differential Equations Using Matlab, read Chapter 9 while seated at the computer.
  • On pages 135-138 of Ordinary Differential Equations Using Matlab, perform the following tasks:
    • Do exercise #2. Use the Page Setup command on the File menu to arrange the figure on the top of the page, leaving whitespace below the figure for you to include the mathematics you used to change the single third order ode into a system of three first order odes. You plot should contain a single graph, the solution of the original third order ode.
    • Do exercise #6. Follow the directions exactly, including the direction about adjusting the time interval on each subsequent plot. Produce at least four plots, a sequence that clearly shows what is meant by a low-pass filter. Arrange your plots using multigraf and in the whitespace at the bottom of your printout, discuss your findings.
    • Do exercise #7. Do parts (a) and (b), then use the Page Setup to arrange your plot at the top of the printed page. Use the whitespace at the bottom half of the page to show your work and explain your results. Do part (c) on a separate printout, arranging and displaying your work as you did in parts (a) and (b).
    • Do exercise #10. In part (a), obtain a printout, arrange at the top of the page using Page Setup and print, and in the whitespace below, copy the commands used at Matlab's command prompt to produce your plot. Follow these same directions in parts (b) and (c). In addition, in the whitespace below your plot on part (c), answer the question posed in part (d).
• Assignment #15
  • Read Section 7.1 in Differential Equations with Boundary Value Problems.
  • On pages 282-283, do exercises #2, 4, 6, 10, 26, 28, 34, 37, 43, and 44. All work should be done by hand, following directions in the text exactly.
• Assignment #16
  • Read Section 7.2 in Differential Equations with Boundary Value Problems.
  • On page 291, do exercises #2, 6, 9, 10, 14, 18, and 22. All work should be done by hand, following any directions exactly.
• Assignment #17
  • Read Section 7.3 in Differential Equations with Boundary Value Problems.
  • Chapter 11 of Ordinary Differential Equations Using Matlab is a good supplementary read for this material. At a minimum, read pages 169-172.
  • On page 299 in Differential Equations with Boundary Value Problems, do exercises #3, 5, 17, 19, and 21. All work should be done by hand, following any directions exactly.
  • On page 300 in Differential Equations with Boundary Value Problems, in exercises #27, 33, and 35, state the system of equations being solved on your homework paper, then state the augmented matrix for the system. Enter the augmented matrix into Matlab, use rref to place the augmented matrix in reduced row echelon form and write the result on your homework, then state the solution of the system in parametric form using vectors.
• Assignment #18
  • Read Section 7.4 in Differential Equations with Boundary Value Problems.
  • Chapter 11 of Ordinary Differential Equations Using Matlab is a good supplementary read for this material. At a minimum, read pages 175-176 on the nullspace of a matrix.
  • On page 307 in Differential Equations with Boundary Value Problems, exercises #11 and 13,state the matrix A on your homework paper, then enter the matrix into Matlab and reduce it with Matlab's rref command. Write the reduced form of the given matrix on your homework and circle the pivots in red. State the number of "free" variables, then write the solution of Ax = 0 in parametric form. State the number of "special" vectors in the parametric solution.
  • On page 307 Differential Equations with Boundary Value Problems, exercises #19 and 21, write the given system in matrix-vector form Ax = b on your homework paper. Enter the augmented matrix for the system into Matlab and reduce it with Matlab's rref command. State the solution of the system in the form x = p + v, where p is a particular solution and v is given in parametric form. Show that v is in the nullspace of matrix A by showing directly that Av = 0. You may do this in Matlab with the Symbolic Toolbox. For example, you can check that sv₁ + tv₂ is in the nullspace of matrix A in Example 4.16 on page 304 in the following manner:

            >> syms s t
    	       >> A=[0 1 3 2 2;1 2 3 5 7;2 4 6 9 15];
    	       >> v1=[3; -3; 1; 0; 0]; v2=[-4; -4; -; 1; 1];
    	       >> v1=[3; -3; 1; 0; 0]; v2=[-4; -4; 0; 1; 1];
    	       >> A*(s*v1+t*v2)
    	       ans =
    	        0
    	        0
    	        0
            
    

  • On page 307 Differential Equations with Boundary Value Problems, do exercises #23, 27, and 29. State the given matrix on your homework, then use Matlab's rref command to reduce the given matrix and state the result on your homework. State the solution of Ax = 0 in parametric form, which describes the nullspace of the given matrix A. Provide a geometric description of the nullpace and sketch your answer in the plane or 3-space as required.
• Assignment #19
  • Read Section 7.5 in Differential Equations with Boundary Value Problems.
  • Chapter 11 of Ordinary Differential Equations Using Matlab is a good supplementary read for this material. At a minimum, read pages 176-179 on linear independence, dependence, and the basis for the nullspace.
  • On page 316 in Differential Equations with Boundary Value Problems, do exercises #6 and 7. Set up the appropriate augmented matrix on your homework paper, reduce it in Matlab, and report the resulting reduced row echelon form of the matrix on your homework paper. Then answer the question exactly as it is posed in the text.
  • On page 317 Differential Equations with Boundary Value Problems, do exercises #21 and 22. In each case, set up the apropriate matrix, reduce it in Matlab, then answer the question exactly as it is posed in the text.
  • On page 317 Differential Equations with Boundary Value Problems, do exercises #29 and 31. In each case, set up the appropriate matrix on your homework, reduce it in Matlab and report the result on your homework, then pick off the "special vectors" and report a basis for the nullspace of the given matrix.
  • On page 317 Differential Equations with Boundary Value Problems, do exercise #35. Set up the appropriate matrix on your homework, reduce it in Matlab and report the result on your homework, then pick off the "special vectors" and report a basis for the span of the given set of vectors.
  • On page 317 Differential Equations with Boundary Value Problems, do exercise #41 and 44. Set up the appropriate matrix on your homework, reduce it in Matlab and report the result on your homework, then pick off the "special vectors" and report a basis for the span of the given set of vectors. Draw an accurate sketch of the span of the given set of vectors, either in two space or three space, as the problem requires.
• Assignment #20
  • Read Section 7.6 in Differential Equations with Boundary Value Problems.
  • Chapter 11 of Ordinary Differential Equations Using Matlab is a good supplementary read for this material. At a minimum, read about the inverse on the bottom of page 174 and top of page 175.
  • On page 322 in Differential Equations with Boundary Value Problems, do exercises #23, 25, and 27. Set up the appropriate augmented matrix [A I] on your homework paper, reduce it in Matlab, then report the resulting reduced row echelon form of the matrix on your homework paper. State whether the matrix is singular or nonsingular. If the matrix is nonsingular, record the inverse of the given mantrix on your homework paper. You might find format rat useful here.
• Assignment #21
  • Read Section 7.7 in Differential Equations with Boundary Value Problems.
  • Chapter 11 of Ordinary Differential Equations Using Matlab is a good supplementary read for this material. At a minimum, read about the determinant and its properties on pages 173-174.
  • On page 329 in Differential Equations with Boundary Value Problems, do exercises #7 and 11. Use hand calculations only to place the matrix in row echelon form (upper triangular), then report the determinant of the matrix. Use Matlab's det command to check your result.
  • On page 329 in Differential Equations with Boundary Value Problems, use hand calculations, expanding across a row or column to determine the determinant in exercises #27 and 29. Secondly, use Matlab's rref to place each of the matrices in reduced row echelon form. On your paper, record the original matrix and its reduced row echelon form. In each case, state a basis for the nullspace of the matrix, state whether the columns of the matrix are independent or dependent, then state the determinant of the matrix.
• Assignment #22
  • Read Section 8.3 in Differential Equations with Boundary Value Problems.
  • Chapter 13 of Ordinary Differential Equations Using Matlab is a good supplementary read for this material. At a minimum, read about nullclines and equilibrium points on pages 212-215.
  • On page 352 in Differential Equations with Boundary Value Problems, perform each of the following tasks for exercises #2, 3, and 6:
    • Use hand calculations to identify the nullclines. Plot the nullclines on a sheet of graph paper.
    • Use hand calculations to find the equilibrium points. Plot these on your graph paper.
    • Use pplane to sketch the nullclines and equilibrium points. Obtain a printout.
  • On page 352-353 in Differential Equations with Boundary Value Problems, perform each of the following tasks for exercise #11:
    • Use hand calculations to answer parts (a) and (b).
    • Use pplane to answer part (c). Obtain two printouts. The first should be a solution of the system defined in equation (3.7) on page 350, using initial conditions F(0) = 40 and S(0) = 20. The second should be a solution of the system defined in equation (3.10) on page 353 with initial conditions x(0) = 40 and y(0) = 20. Use good prose to describe the effect of modifying the model and explain how this agrees with your findings in parts (a) and (b).
  • On page 353 in Differential Equations with Boundary Value Problems, perform each of the following tasks for exercise #12:
    • Use hand calculations to answer parta (a). Place your findings on graph paper.
    • Use pplane to answer part (b).
    • Use pplane to answer part (c). Obtain two printouts. The first should be a "phase portrait" of the system defined in equation (3.7) on page 350. The second should be a "phase portrait" of the system defined in equation (3.11) on page 353. Use good prose to describe the differences in the "phase portraits."
• Assignment #23
  • Read Section 8.4 in Differential Equations with Boundary Value Problems.
  • On page 359-361 in Differential Equations with Boundary Value Problems, do exercises #15, 16, 19, 22, 25, and 29.
• Assignment #24
  • Read Section 8.5 in Differential Equations with Boundary Value Problems.
  • On page 368 in Differential Equations with Boundary Value Problems, do exercises #4, 10, 16, 23, and 26. Use pencil and paper calculations only.
  • On page 369 in Differential Equations with Boundary Value Problems, do exercises #28 and 29. Perform each of the following tasks:
    • Sketch the tank picture on your homework paper. Label everything very precisely.
    • Set up a system of three first order differential equations in the unknowns x_A, x_B, and x_C. Use the initial salt content in each tank to set up the initial conditions for the system.
    • Write a function ODE file, a script to drive the solver, and plot the salt content in each tank over time (use ode45).
    • Submit printouts of the plot, your script, and the function ODE file with your homework assignment.
• Assignment #25
  • Read Section 9.1 in Differential Equations with Boundary Value Problems.
  • On page 376 in Differential Equations with Boundary Value Problems, do exercises #1, 12, 17, 20, and 25. Use pencil and paper calculations only.
  • On page 377-378 in Differential Equations with Boundary Value Problems, do exercises #49, 50, and 51. Use Matlab's eig command to find the eigenvalues of the given matrices. When asked for the determinant, use Matlab's det command.
  • In Ordinary Differential Equations Using Matlab, read pages 181-186.
  • In Ordinary Differential Equations Using Matlab, page 207, do exercises #4 and 8. Use Page Setup to center your plot at the top of the page. In the remaining whitespace on the page, write in the results of the other commands requested in the exercises.
  • In Ordinary Differential Equations Using Matlab, page 208, do exercises #9, 10, 11, and 12. Record the commands used and their results on your homework.
• Assignment #26
  • Read Section 9.2 in Differential Equations with Boundary Value Problems.
  • On page 390 in Differential Equations with Boundary Value Problems, perform each of the following tasks for exercises #11 and 25:
    1. Sketch the solution of the initial value problem in pplane. Locate and classify the equilibirum point of the system using pplane.
    2. Obtain an x and y versus t plot in pplane. Include printouts of these results with your homework papers.
    3. Solve the initial value problem using hand calculations only.
    4. Using your solution found in part (3), use Matlab to provide a plot of y versus x, and a second plot of x and y versus t. Use a domain such as [-4T_c, 4T_c], where T_c is a compromise of the time constant(s) for the system. Obtain printouts of these results and compare them with your pplane results.
  • On page 390 in Differential Equations with Boundary Value Problems, use hand calculations only to compute the solution of the initial value problems in exercises #35 and 37.
  • On pages 390-391 in Differential Equations with Boundary Value Problems, do exercises #58 and 61. Use Matlab to build the required plots. Obtain printouts and submit them with your homework papers.
• Assignment #27
  • Read Section 9.3 in Differential Equations with Boundary Value Problems.
  • On page 401 in Differential Equations with Boundary Value Problems, do exercises #10, 11, and 12 on graph paper.
  • On page 402 in Differential Equations with Boundary Value Problems, perform each of the following tasks for exercises #18, 21, and 23:
    1. Follow the given directions for each exercise, sketching your results on graph paper. Hand calculations only. Show your work.
    2. Use pplane to verify your result. Obtain a printout for each exercise.
• Assignment #28
  • On a piece of standard sized posterboard, sketch the trace-determinant plane. Determine all possible classes of equilibrium points in the trace-determinant plane and use pplane to sketch particular examples of each, pasting each example in the appropriate place in the trace determinant plane. Argue with your classmates over the possible number of distinct cases. Some reading sources for this exercise include:
    1. Differential Equations with Boundary Value Problems, chapter 9, section 4, pages 402-405, and
    2. Ordinary Differential Equations Using Matlab, exercise #17, pages 230-231.
• Assignment #29
  • Read Section 10.1 in Differential Equations with Boundary Value Problems.
  • On page 468 in Differential Equations with Boundary Value Problems, perform each of the following tasks for exercise #3:
    1. Sketch the nullclines on graph paper, the x-nullclines in red, the y-nullclines in blue. Label each of the equilibrium points with their coordinates. If this requires solving systems, show your algebraic work.
    2. Calculate the Jacobian by hand.
    3. Use the Jacobian to classify each equilibrium point, showing your work in each case, identifying T, D, and T² - 4D in each case.
    4. Use pplane to obtain a phase portrait of the system, including nullclines and equilibrium points labeled with their coordinates.
• Assignment #30
  • Read Section 10.2 in Differential Equations with Boundary Value Problems.
  • On page 474 of Differential Equations with Boundary Value Problems, use hand calculations only to provide the analysis requested in exercise #5.
  • On page 474 of Differential Equations with Boundary Value Problems, use the Symbolic Toolbox to provide the analysis requested in exercise #7.
  • On page 475 of Differential Equations with Boundary Value Problems, you may use the results developed in the narrative of section 10.2 to find and analyze the equilibrium points in exercises #17, 18, and 19. However, use ode45 to plot the requested solution. Include a printout with your homework papers.
• Assignment #31
  • Read Section 10.3 in Differential Equations with Boundary Value Problems.
  • On page 480 of Differential Equations with Boundary Value Problems, perform each of the tasks (by hand) delineated in exercise #9. Put your result on graph paper, then use pplane to duplicate your results. Obtain a prinout. Note: There is a bug in the pplane software that incorrectly draws the arrows associated with the nullclines. It is recommended that you use the Java version of pplane at http://math.rice.edu/~dfield/dfpp.html.
  • On page 480 of Differential Equations with Boundary Value Problems, duplicate the tasks delineated above for exercise #9 and write up a solution of exercise #13.
  • On page 480 of Differential Equations with Boundary Value Problems, use hand calculations only to answer the questions posed in exercise #17. Once you've completed that task, use pplane to provide two examples, one with ab<1 and a second with ab>1. Provide nullclines and equilibrium points and enough trajectories so that the behavior of the system is clear.
• Assignment #32
  • Read Section 10.4 in Differential Equations with Boundary Value Problems.
  • On pages 488-40 of Differential Equations with Boundary Value Problems, perform each of the tasks (by hand) delineated in exercises #1, 5, 7, 11, 17, and 22, then confirm your results experimentally with pplane. Obtain a printout(s) for each exercise and include it with your homework packet.