Homework due on Monday, Sept. 17:
4.1: 1,7,12,23,26
4.2: 13,23,26,31,35
4.3: 3,7,9,11,13,19,24
Homework due Monday, Sept. 24:
4.3: 37,38,41,45,47,49--54,65
4.4: 3,7,11,13
Homework due Monday, Oct. 1:
4.4: 19,21,27,33,37,39
4.4: 49
4.5: 1,5,9
Our first exam will take place on Monday, Oct. 8, and cover sections 4.1--4.6. You won't be allowed to use any notes or calculators. You don't need to memorize the formula for the Laplace transform of a periodic function, nor the 3 formulas on p. 222 which start on line 6. But you will need to know the following formulas from Appendix III (located after Chapter 20): 1--8, 11--13, 16--19, 37--39, 52--57.
Some suggested review problems for the exam are p. 236: 1,7--25 odd,29,31,33,37,39 plus 4.5: 3
Homework due Monday, Oct. 8: 4.6: 1,7,9
Homework due Monday, Oct. 15:
3.9: 13,15
12.1: 7,8,17
Homework due Monday, Oct. 22:
12.2: 1,5,17, and for 1, 5 sketch the graphs of both f and S[f] on the closed interval [-2 pi, 3 pi].
Homework due Monday, Oct. 29:
12.3: 2,3,8,13,23,29
Homework due Monday, Nov. 5:
13.1: 4,11
Homework due Monday, Nov.12:
13.3: 1,4. Find lim_{t \to + infinity} u(x,t) for both problems. In case L = pi and k =1 in problem 1, use the first term of the series for u to approximate the temperature at the center of the bar when t = 1. For the second problem, use the first 2 terms of the series to approximate the temperature at the center when t=1 (and k=1).
13.4: 1,5
Homework due Monday, Nov. 19:
Sketch the graphs of u(x,1) and u(x,3) for x in [0,12] where u(x,t) is the solution of u_{tt} = 9 u_{xx}, 0 < x < 12, t > 0, with u(0,t) = u(12,t) = 0 for t >0 and initial conditions u(x,0) = f(x) and u_t(x,0) =0 for 0 <= x <= 12 with f(x) corresponding to plucking at x=4 with amplitude 4. (You don't need to compute a series for u(x,t).)
13.5: 7,15 (You may use the solutions on p. 710.)
PLUS
(1) Solve the eigenfunction problem X'' + lambda X = 0 for 0 < x < L with X(0) = X'(L) = 0. Consider all possible cases. Answer: sin [(2n+1) pi x/ (2L)], n =0, 1, 2,....
(2) Find the steady state temperature for the heat problem u_t = 4 u_{xx}, 0 < x < 10, t > 0, with BC u(0,t) = 200 and u(10,t)= 300 for t > 0, and with IC u(x,0) =f(x) for 0 < x < 10. (Hint: subtract an appropriate linear function of x from u(x,t) to reduce to the case of zero BC.)
EXAM 2 will be held on Monday, Nov. 26
The answer to 14.1: 4 is u(r, theta)= (1/2)a_0 + sum_{n=1}^{infinity} r^n(a_n cos n theta + b_n sin n theta) where f(theta) = theta = (1/2)a_0 + sum_{n=1}^{infinity} (a_n cos n theta + b_n sin n theta). It follows that all a_n are 0 (since the function f(theta) = theta is odd) and b_n = (2/n) (-1)^{n+1}. The formula for b_n is obtained by using integration by parts. You don't need to hand-in the solution for 14.1:4.
Homework due Monday, Dec. 10:
8.8: 5,13,19
8.6: 7,21,43
Our final exam date is Tuesday, Dec. 18, 12:00-3:00 PM, and the room is our regular one: SEC 212.