Math 33B, Section 1C and 1D: Differential Equations, Fall 2007 (RJ Miech ) Teaching Assistant: Eric Radke (email)   Course Mechanics for Math 33B, Lecture 1 Announcements: Prof. Miech would like you to turn in the homework that was assigned on 11-21, 11-26, 12-03, and 12-05 on Friday, 12-07. There will be a review for the final in class on 12-07.   Homework: Assigned 9-28:  Pg. 25-26:  Problems 3, 4, 21, 23, 33 Assigned 10-01:  Pg. 35-36:  Problems 5, 9, 11, 19, 21, 34, 35 Assigned 10-03:  Pg. 55:  Problems 1, 3, 7, 13a, 17, 19 Assigned 10-05:  Pg. 61:  Problems 1, 5, 7a Assigned 10-08:  Pg. 75:  Problems 9, 11, 13, 15, 17, 19, 21, 31, 33 Assigned 10-10:  Pg. 100:  Problems 1, 5, 11, 13.  Do them "by hand;" don't use the computer for sketching.  Also, please note that the answer in the back of the book for Problem 5 is wrong. Assigned 10-15:  Pg. 156:  Problems 1, 3, 5, 7, 9, 11, 13, 15 Assigned 10-17:  Pg. 172:  Problems 1, 2, 3, 4, 5, 6 Assigned 10-19:  Pg. 177:  Problems 1, 3, 5, 7 Assigned 10-22:  Pg. 376:  Problems 1, 3, 5 (no assignment 10-24) Assigned 10-29:  Pg. 389:  Problems 1, 3, 5, 7, 9 Assigned 10-31:  Pg. 390:  Problems 17, 19, 21, 23, 25, 27 Assigned 11-02:  Pg. 390:  Problems 29, 31, 33, 35, 37, 39 Assigned 11-05:  Pg. 389:  Sketch the half-line solutions of the differential equations in Problems 1, 3, and 5.  Then sketch a rough approximation of a solution in each region determined by the half-line solutions.  (You can go to PPlane at the link below to see some direction fields for 2 x 2 systems.) Assigned 11-07:  Pg. 401:  Problems 13, 17 Assigned 11-09:  Pg. 414:  Problems 7, 9, 21 Assigned 11-14:  Using software, find the solutions of Problems 37, 40, and 41 on Pg. 415.  The solution can be expressed as a matrix, as was done in class.  In addition, for each of these problems, assuming the solution is [x(t) y(t) z(t)]^T, find the numerical value of                           [x(0) y(0) z(0)]^T.  NOTE:  To get [x(t) y(t) z(t)]^T, take c_1 = 1, c_2 = c_3 = 0 from your general solution, i.e. just use exactly the first column of your solution matrix. Assigned 11-16:  Pg. 542:  Problems 1, 3, 5, 7, 9, 11 Assigned 11-19:  Pg. 543:  Problems 13, 27, 29, 31, 35 Assigned 11-21:  Pg. 554:  Problems 11, 13, 15, 17, 19.  In problems 15, 17, and 19 you are asked to show that x_0 is an "ordinary" point of the differential equation y''(x) + p(x)y'(x) + q(x)y(x) = 0.  Then p(x) and q(x) are infinitely differentiable at x_0 = 0.  For problems 15, 17, and 19 this is obvious, so you can ignore that part of the problem.  In short, just find the series expansions for the solution.  In addition, the solutions to 15, 17, and 19 are given in "closed" form; that is, they are expressed as combinations of sines, cosines, and exponentials.  You need not do this; just find the series solutions. Assigned 11-26:  Pg. 554:  Problems 21, 23 (no assignment 11-28) Assigned 12-03:  1.  Solve dy/dx = x^2/[y(1+x^4)].  Hint:  Use the power series expansion for 1/(1+x^4) to get a solution valid for |x|<1.  2.  Solve y''(x)+y'(x)/(1+x) = 1 + x.  Hint:  Set u(x) = y'(x).  3.  Solve y'(x) + 2y(x) = x^5.  4.  Find the power series solution of  y'(x) + x^3y(x) = x^2. Assigned 12-05:  1.  What condition must hold if y''(x)+ay'(x) +by(x)=0 is to be a bounded function?  2.  Suppose that y''(x)+ay'(x) +by(x)=0 has the general solution y(x) = c_1e^(-4x)cos(2x)+c_2e^(-4x)sin(2x).  What are the associated eigenvalues for this differential equation?  3.  Let y''(x)+p(x)y'(x)+q(x)y(x)=0  (Eqn 3).  State general conditions on p(x) and q(x) that will guarantee there is a function, defined for all x, which is a solution for (Eqn 3).  4.  Find the general solution of y''(x) +y'(x)/(1+x)=1+x.  Hint: set u(x) = y'(x).  5.  Find the general solution of                           y''(x) +6y'(x)+9y(x)=0.   My Sections: 1C: Tuesdays, 9:00 - 9:50 AM, Royce 164 1D: Thursdays, 9:00 - 9:50 AM, Royce 162   Office Hours Info: Office: Math Sciences 2963 Office Hours: Thursdays, 11:00 - 12:00 in the Student Math Center , Math Sciences 3974 Thursdays, 12:00 - 1:00 in my office   Useful Links: dfield and pplane, a useful tool created by the author of the textbook (Polking) for drawing direction fields and phase planes.   Other Math 33B Lecture 1 TA's: Miyoun Jung.  Office Hours:  Mondays, 11:00-12:00 in the SMC (MS 3974); and Thursdays, 10:00-11:00 in MS 7630.