Math 23 "Differential Equations"

Fall 2012

Section taught by Sergey Melikhov

Announcements

Final Exam: Friday 11/16, 11:30am-2:30pm in Kemeny Hall, Room 008

 Lecture Dates Sections in Textbook Homework Problems Deadlines for Homework Wednesday 10/24 7.5 9, 11, 12, 14 (cont'd) Due on Wednesday 10/31. Thursday 10/25 x-hour No class on Friday 10/26 Homecoming weekend 7.6 3(a), 5(a), 7 (8th edition: only solve and express in terms of real-valued functions in problems 3, 5) Monday 10/29 7.8 2(c), 3(c), 5 (8th edition: only solve the system in problems 2, 3) Due on Wednesday 11/7. Wednesday 10/31 7.8 10.1 1, 5, 10, 14 Friday 11/2 10.5 3, 5 10.7, 10.B Monday 11/5 10.7 Solve #1(a,b) using the method of D'Alembert (discussed in class; a sketch can be found in Problems 13, 14, 16 and 20 in Section 10.7). Due on Monday 11/12. Same numbers in the 8th edition. Wednesday 11/7 10.7 Solve #1(a) using Fourier series. 10.2 14, 15 Thursday 11/8 x-hour No class on Friday 11/9 10.3 6 10.4 16, 17 10.5 9, 10 Monday 11/12 No class on Wednesday 11/14 Pre-examination break 11/14-11/15 10.6 9(a) This assignment will not be collected or graded, but similar problems may appear on the final. 10.7 5(a)

Final: Friday 11/16,

11:30am-2:30pm in Kemeny Hall, Room 008

Final Exams End on Tuesday 11/20

OLD STUFF

 Lecture Dates Sections in Textbook Homework Problems Deadlines for Homework Monday 09/10 1.1 #7. Write down a differential equation of the form dy/dt=ay+b whose all solutions approach y=3 as t→∞. #12. Draw a direction field for y'=-y(5-y). Based on the direction field, determine the behavior of y as t→∞. If this behavior depends on the initial value of y at t=0, describe this dependency. Due on Wednesday 9/19 1.3 Wednesday 9/12 1.2 #8. Consider a population p of field mice that grows at a rate proportional to the current population, so that dp/dt=rp. (a) Find the rate constant r if the population doubles in 30 days. (b) Find r if the population doubles in N days. 2.1 Friday 9/14 1.3 (cont'd) #12. Verify that y1(t)=t-2 and y2(t)=t-2lnt are solutions of t2y''+5ty'+4y=0 for t>0. 2.1 (cont'dite) #15. Find the solution of the initial value problem ty'+2y=t2-t+1, y(1)=1/2, for t>0. #17. Find the solution of the initial value problem y'-2y=e2t, y(0)=2. #33. Show that if a and λ are positive constants, and b is any real number, then every solution of the equation y'+ay=be-λt has the property that y→0 as t→∞. Hint. Consider the cases a=λ and a≠λ separately. 2.2 #3. Solve the equation y'+y2sinx=0. #8. Solve the equation dy/dx=x2/(1+y2). Monday 9/17 2.4 #3. Without solving the initial value problem y'+(tant)y=sint, y(π)=0, determine an interval in which the solution is certain to exist. #14. Solve the initial value problem y'=2ty2, y(0)=y0, and determine how the interval in which the solution exists depends on the initial value y0. #25. Let y=y1(t) be a solution of y'+p(t)y=0 and let y=y2(t) be a solution of y'+p(t)y=g(t). Show that y=y1(t)+y2(t) is also a solution of the latter equation. #33. Solve the initial value problem y'+p(t)y=0, y(0)=1, where p(t)=2 for 0≤t≤1 and p(t)=1 for t>1. Due on Wednesday 9/26. All problem numbers also work for the 8th edition. Wednesday 9/19 2.4 (cont'd) 2.3 Thursday 9/20 x-hour 2.3 (cont'd) 2, 4 and 8(a,b) Friday 9/21 2.5 We draw the entire graph and phase line, in contrast to the first quadrant / positive ray figures in the book. 3, 13 and 15(a) 2.4 (cont'd) 28 Monday 9/24 2.6 2, 14, 28 Due on Wednesday 10/3. Same problem numbers in the corresponding sections in the 8th edition. Wednesday 9/26 2.6 24, and the following problem: #24α. Use #24 to solve the equation (3y2+4xy)+(4xy+3x2)y'=0. 3.1 11, 12 4.1, 4.2 Thursday 9/27 x-hour 3.3: Complex Roots (=3.4 in the 8th edition) 1, 5, 11, 19 Friday 9/28 3.4: Repeated Roots (=3.5 in the 8th edition) 2, 13, 14, 23, 25 Monday 10/1 3.5, 3.8 (=3.6, 3.9 in the 8th edition) Due on Wednesday 10/10

Midterm 1: Tuesday 10/2,

6-8pm in Carpenter Hall, Room 013

 Lecture Dates Sections in Textbook Homework Problems Deadlines for Homework Monday 10/1 3.5, 3.8 (=3.6, 3.9 in the 8th edition) Due on Wednesday 10/10. Same problem numbers in the corresponding sections in the 8th edition. Wednesday 10/3 3.5: Undetermined Coefficients (=3.6 in the 8th edition) 1, 2, 13, 14, 16, 18, 19(a) 3.8 (=3.9 in the 8th edition) Friday 10/5 3.7: Vibrations (=3.8 in the 8th edition) 6 3.8: Forced Vibrations (=3.9 in the 8th edition) 9, 11 Monday 10/8 3.8: Forced Vibrations (=3.9 in the 8th edition) Due on Wednesday 10/17. 7.3 3 Wednesday 10/10 7.3 8, 29 (=7, 28 in the 8th edition) In #29 don't assume anything about det A. Thursday 10/11 x-hour 4.1 8 Friday 10/12 3.2 11 4.1 2, 6 7.1 5, 9 7.4 5 Monday 10/15 4.2 2, 13, 14, 23, 25 Due on Wednesday 10/24. 7.4 Wednesday 10/17 3.2 23, 27 (22, 26 in the 8th edition) 7.4 6 Thursday 10/18 x-hour 3.2 (3.3 in the 8th edition) 35, 39 (=21, 25 in the 8th edition) Friday 10/19 3.2 19 3.6 (=3.7 in the 8th edition) 10, 15 Monday 10/22 7.3 19, 25 (=18, 24 in the 8th edition) Due on Wednesday 10/31.

Midterm 2: Tuesday 10/23,

6-8pm in Carpenter Hall, Room 013