Math 320 — Homework Assignments

Spring 2010

Homework will be assigned every monday in lecture, and later posted on this page. Homework must be returned in lecture on the following monday. It should be neatly written; separate sheets must be stapled (no paper clips, etc.); your name must appear on each sheet. Homework that does not meet these standards will not be graded.

Due Monday May 3.

In addition, use the method of “guessing” to solve problems 1, 4, 7, 8 from §8.2 in the book. (The method is described briely in the notes, and also in the beginning of §8.2, where it is called the “method of undetermined coefficients.”

Due Monday April 26.

• §7.1: Examples 3 and 4 show you how to turn one or more higher order diffeqs into a system of first order equations. Example 5 shows how to do the opposite.
• §7.1: The two paragraphs at the bottom of page 400 explain (very briefly) what a direction field and what the phase plane portrait of a system of two first order differential equations is.
• §7.4 The paragraphs on page 434, beginning with Solution of Second Order Systems explain how to solve such systems. In the special case that the matrix only has negative eigenvalues Theorem 1 summarizes the result. It is worth knowing the derivation since it also applies to variations on this kind of problem (e.g. if you add friction or a magnetic field), while Theorem 1 doesn't tell you anything about slightly different equations. I will go over this in class on Friday.

• §7.1: 1—6. Which of these equations are linear? Which are linear with constant coeficients? For those equations for which the first order systems you find are linear with constant coeficients, find the matrix A.
• §7.3: 38, 39.
• §7.3: Use Octave or Matlab or the online Math Calculator to solve problem 45. Also find the characteristic polynomial of the matrix in problem 45 (demonstration in lecture on Wednesday.)
• §7.3: Read example 4, and in particular the explaining paragraphs before that example, on page 427. Find the eigenvalues corresponding to the system (22) (page 427) in which k₁=k₂=1, and k₃=K (K is some positive number). For which values of the constant K do you get complex eigenvalues, and for which do you get only real eigenvalues?
• §7.4: 1, 3, 5, 7.

Due Monday April 19.

• §7.3: Read the entire section carefully. It explains how to use the eigenvalues and eigenvectors to compute the general solution to almost any system of linear ordinary diffeqs with constant coeficients.

• §7.3: 3, 6, 9, 11, 19, 22, 24, 25.
  This is not a very long list of problems, but they are all multistep problems so don't underestimate the amount of work needed.

Due Monday April 12.

• §6.1: Read the entire section carefully. It explains what eigenvalues and eigenvectors are, and how to compute them. The Algorithm on page 369 summarizes how you go about finding eigenvalues/vectors; the examples in this section show you the details of the process.
• §6.2: Summary: An n-by-n matrix is diagonalizable if you can find n linearly independent eigenvectors for the matrix. So the question “can the matrix A be diagonalized?” is the same as “can you find n linearly independent eigenvectors for A?” Eigenvectors corresponding to different eigenvalues are always linearly independent.

• §6.1: 6, 10, 14, 18, 19, 23, 29, 32, 33, 34.
• §6.2: 1, 5, 8, 12, 13, 14, 21, 22.

Due Monday March 22.

• §5.2: Linear independence is the concept which tells you when n solutions y₁, …, y_n (of a linear homogeneous equation) are good enough to generate the general solution via y = c₁y₁ + … + c_ny_n. The formal definition is given in the book on page 305. There, and on the following page (306) it is related to the Wronskian. See examples on pages 306, 307 for the use of the Wronskian
  
• §5.2: Page 310 explains how to solve nonhomogenous equations.
• §5.3: If you have forgotten how to solve constant coefficient linear equations (in particular, if you need to be reminded on what to do if there are complex roots) all you need to know is in this section. Examples 4, 5, 7, and 8 (pages 321–323) are typical of what you need to know.
• §5.6: Resonance is an important example of how solutions of a linear differential equation depend on the frequency in the nonhomogeneous term. Think of this chapter as a very long worked example of the theroy explained in math 222, and in the previous sections in the book.

• §5.2: 13, 15, 19. In addition to doing these problems, compute the Wronskian for the given solutions in each case. Finally, which of these equations has constant coefficients?
• §5.2: 21, 24, 31.
• §5.3: 9, 15, 17 — also compute the Wronskians of the solutions you find.
• §5.6: 4, 11, 12 (try to find particular solutions of the form A cosωt + B sinωt, as explained in example 6, page 360/361.)

Due Monday March 15.

• §2.2: Make sure you understand the relations between Figures 2.2.1 and 2.2.2 on page 93, and between figures 2.2.3 and 2.2.5 on page 95.
  
• §2.4: Make sure you understand Euler's method (middle of page 113, until box on page 114.)

• Explain in two or three sentences why the authors of the book call equation (9) (page 95) the “explosion/extinction equation.”
• §2.2: Find the critical points, the phase diagram, and use your phase diagram to determine which of the critical points are stable, for the differential equations in problems 1,2,3,4,7,9,10. You do not have to find the general solution to these equations.
  In each of these problems, let x(t) be the solution for which x(0) = 0.73. Compute the limit of x(t) as t goes to +∞. You can do this using the phase diagram you found, and you do not need to compute the general solution of the diffeq.
• §2.4: Do problem 4. You will probably need a calculator. Do all computations in at least three decimal places.
• Go to the the logistic equation solver at the Virtual Math Museum. The equation being solved is y' = ay(b-y) with a=1 and b=5. Select "Euler's method" and "Connect dots", and set the step size to 2.0 (instead of the default value 0.1). Plot a few solutions, choosing initial values somewhere between 0 and 5. What is most noticably wrong with the solutions produced by Euler's method at step size h=2.0? (Do the solutions converge as time goes to ∞?)

Due Monday March 8.

• Two solutions are drawn in the picture: let's call them y₁ and y₂. Are they equal at x=-3, ie. is y₁(-3) = y₂(-3)? (you should answer this question without solving the equation. If you use a theorem, say which one it is, and quote the page in the book where you can find the theorem.)
  
• Find the general solutions of the diffeq. (You're supposed to know how to do this. It's a linear first order equation, and it can be solved by multiplying with an integrating factor. If you have forgotten how this works, read §1.5 of the book, in particular example 1 on page 50, which is very similar to this problem. If you get stuck on an integral you could try integrals.com.)
• Which values of the constant in your general solution correspond to y₁ and y₂? How much do y₁(-3) and y₂(-3) really differ? (From the picture you can see that y₁(3)≈1.7 and y₂(3)≈-2.2.)

Due Monday March 1.

“Due” Monday February 22.

Due Monday February 15.

Due Monday February 8.

Due Monday February 1.

Due Monday January 25.