Homework numbers are from the 9th edition of Elementary Differential Equations and Boundary Value Problems, by William E. Boyce and Richard C. DiPrima. If you have the 8th edition, you should have most of the problems: see this page for more information.
Homework 7 — due Friday, 6/7
To turn in:
6.3: 6, 12
6.4: 6a, 5a, 4a (these are listed in rough order of difficulty)
This week's homework problems are here. Here are answers for the additional problems so you can check your answers.
Homework 5 — due Friday, 5/17The homework problems for this week are listed here.
Homework 4 — due Friday, 5/10
To turn in:
3.5: 5, 28***, 29**, 12*, 15*
3.5: 18*; 3.7: 3, 4 (will be on homework 5 instead)
*Important: Wait on these problems until we talk about multiplying trial solutions by t.
**Note on 3.5 #29: To plot the graphs of these piecewise defined functions, you can use a graphing calculator or an online tool like this one or Wolfram Alpha to plot each piece of the function separately (the part 0 ≤ t ≤ π and the part t ≥ π).
*** Hint on 3.5, #28: You don't have to tackle the entire sum on the right hand side at once. Here's a fact that might help: suppose y1(t) and y2(t) are two functions, and say that when you plug y1(t) into y'' + λ2y you get the function g1(t) and when you plug y2(t) into y'' + λ2y, you get g2(t):
y1(t)'' + λ2y1(t) = g1(t)
y2(t)'' + λ2y2(t) = g2(t)
Now if you plug in y1(t) + y2(t), you should check that you get
(y1(t) + y2(t))'' + λ2(y1(t) + y2(t)) = g1(t) + g2(t)
How can we use this fact? Well, let's imagine N = 2 in the problem for a second, so the the right-hand side of our nonhomogeneous equation consists of two functions added together, a1 sin πt + a2 sin 2πt. Instead of solving the original equation, we can solve the two simpler equations
y1(t)'' + λ2y1(t) = a1 sin πt
y2(t)'' + λ2y2(t) = a2 sin 2πt
and then add together y1 and y2 to get a solution to the original equation, using the fact we found above.
Homework 3 — due Friday, 5/3
To turn in:
2.7: 20
3.1: 23, 26(a,b,d)
3.4: 11**, 32*
3.3: 19**, 38***
* Note on 3.4, #32: In the final step of this problem, you will encounter an integral that can't be evaluated. In your final answer, you can leave this integral as is.
** Note on 3.3, #19, 3.4 #11: Describing the solution's behavior for increasing t means: what does the solution do as t goes to infinity? Does it have a limit, or does it grow without bound? Write a sentence or two to explain what the solution is doing and show any necessary calculations to back up your statement.
*** Note on 3.3 #38: You don't need the information from section 3.3 (which we won't cover in class until Wednesday) to do this problem.
Extra problems for practice (not to be turned in; will not be graded):
2.7: 15
3.1: 20, 22, 27
3.4: 29, 39
3.3: 5, 24
To turn in:
2.1: 24(b)*
2.3: 4, 13, 19, 21(a,b), 26, 27(a)
Note on 2.1, problem 24(b): Don't worry about finding the critical value — you can just solve the equation...
Homework 1 — due Monday, 4/8
To turn in:
1.3: 14
2.2: 22, 26, 34
2.1: 2(a-b), 7c, 31