Math 285 Homework, Spring 2010

Beginning of Announcements.

Your homework must start with a cover sheet. Print the sheet for either Section F1 (2pm) or Section G1 (3pm).
All homework must be stapled. All answers that are formulas must be boxed, in your solutions.
Tips on using Wolfram Alpha are now available. Use Wolfram Alpha to investigate homework problems, check your answers, and more.

End of Announcements.

Homework must be turned in by the end of class on the due date. Late homework will be accepted only if you have made prior arrangement with me. Exceptions will be made, for example, in case of serious illness or family emergency.

Form a study group to discuss the homework problems. But write up your own solutions: you will only learn properly if you think the problems through for yourself.

Try asking and answering questions at the Google discussion group. Please title your post with the number of the relevant homework problem (such as "3.2 #17"). The professor and TAs will sometimes respond to posts, but the intention of the group is for students to discuss the homework with each other.

The median scores on the Homeworks so far are: 9, 8.5, 9, 9.5, 9, 9.5, 9.5, 9 9, 8, 9, (all out of 10)
The median score on Test 1 is: 60/75
The median score on Test 2 is: 83/100

Grading scheme: on each homework out of 10 points, the grade will be assigned as follows:

I will drop your 2 lowest homework scores.

Tips on using Wolfram Alpha are now available. Use Wolfram Alpha to investigate homework problems, check your answers, and more.

Homework # 12
9.6 Download and print Homework 12. Due Friday 7 May by 4pm, to my office 376 Altgeld.
Homework # 11
9.5 and 9.6 Download and print Homework 11. Due Friday 30 April in class.
Homework # 10
9.3 #17 - do it graphically, representing the integrals as areas. Then do two additional parts:
(a)' - Suppose that f is an even function. Show that int-aa f(t) dt = 2 int0a f(t) dt.
(b)' - Suppose that f is an odd function. Show that int-aa f(t) dt = 0.

Problem A. Do part (iii) below, about the product of two odd functions. You are not required to do parts (i) and (ii).
(i) (even function)(even function)=even function
(ii) (even function)(odd function)=odd function
(iii) (odd function)(odd function)=even function
Note.These rules are not the same as for multiplying odd and even integers. Rather, they are analogous to the rules for adding odd and even integers. The essential reason is that raising x to an odd power gives an odd function, and raising x to an even power gives an even function, and multiplying the functions means that you add the exponents.

Problem B. Use 9.3 #17 and Problem A parts (a)' and (b)' to show:
(i) if f is an even function then int-pipi f(t) cos(nt) dt = 2 int0pi f(t) cos(nt) dt, and int-pipi f(t) sin(nt) dt = 0.
(ii) if f is an odd function then ...[your task is to state some analogous results, but you are not required to prove them].

9.3 #5 - do not compute the series. Instead, your task is to use the sine series given in the answer at the back of the book to find two series for pi (the two series must be different). Write out 9 terms in each series for pi, to make the pattern clear.
Hint. You can use that the coefficients bn in the sine series are: 2/(n pi) when n=1,7,13,..., and -4/(n pi) when n=3,9,15,..., and 2/(n pi) when n=5,11,17,...

9.4 #2. Then also:
(a) Plot the particular solution xp(t) and the forcing function F(t) on the same graph. (Either use your graphing calculator, or use Iode with initial conditions x(0)=0.6,x'(0)=0, for forcing function F(t)=3sign(cos(pi t/2)). The initial condition 0.6 has been chosen to approximately equal xp(0). Make sure you use Runge-Kutta with step size 0.05, to get an accurate solution. Do not use Euler! Use "Plot arbitrary function" to plot the forcing function.)
(b) Estimate the dominant period in the response xp(t), by using your graph. Compute wp=2pi/(dominant period).
(c) Compare your frequency wp with the natural frequency of the system. If they are close, then explain why this has happened, using as evidence the formula for the coefficient an in your Fourier series for xp(t).
[APOLOGY: ignore part (c) of the problem! The dominant period in the graph of xp(t) is 4, so wp=pi/2. The natural frequency is sqrt(10), which is not close to pi/2. I was thinking that the term with n=2 would dominate the solution (because putting n=2 into the formula for an would give a very small denominator), which students could then explain by noting that the n=2 term in F(t) is close to resonance because its frequency 2pi/2=pi=3.14... is very close to the natural frequency sqrt(10)=3.16... My mistake was to forget that in this particular problem, n is odd and so an=0 for all even n, in particular a2=0. So the dominant term in the response turns out to be the term with n=1, corresponding to frequency pi/2 and period 4.]

9.4 #18. Do not use formula (16). Instead, derive your answer from first principles: find the Fourier series for F(t), and substitute it and a Fourier series for xp(t) into the differential equation, and then determine the coefficients in xp(t). (Note your Fourier series for xp(t) will need both sine and cosine terms.)
(a) For n=1,2,3,4,5,6,7, evaluate the amplitude (an2+bn2)1/2 of the coefficients. For which n is the amplitude largest?
(b) Compare that dominant frequency n with the practical resonance frequency wmax of the system (see Sec. 3.6 #27). Briefly discuss your findings.

Due Friday 23 April in class.
Homework # 9
Problem A. Compute the Fourier series of f(t)=4-5sin(3t)+7cos(6t), -pi < t < pi. Hint. Orthogonality of sines and cosines, at the end of the Orthogonality handout.
9.1 #18 - sketch the graph and then evaluate bn. You will get more credit if you find an intelligent way to do this problem. (Optional, for no credit: evaluate an.)
9.1 #21 - and deduce that 1/12+1/22+1/32+1/42+... = pi2/6.
Hint for #21. The 2pi-periodic extension of f(t)=t2 is a piecewise smooth function that is continuous at t=pi (which you should check by sketching its graph; note it has a corner at t=pi). What does the Fourier Convergence Theorem tell you about the value of the Fourier series at t=pi?
9.3 #6 (make sure you sketch the extension over at least two periods)
9.1,9.2 - Iode project on Fourier series. Turn in pages 5-8, containing your answers.

Due Friday 16 April in class.
Lab office hours: Thursday 15 April, 3:30-5:00pm, in Engineering Hall 406B1 lab
Homework # 8
3.6 # 6, 8 (the reason for plotting the response and the forcing function together is to observe that the response lags behind the forcing),
18 (plug into formula (21), then plot C(w), and find the maximizing value of w by calculus),
24 (conclusion: when the forcing is not purely a sine or cosine, there can be more than one resonant forcing frequency due to "overtones" in the forcing),
Hint for #24: cos3(u) = (3/4) cos(u) + (1/4) cos(3u) by 3.5 #43a. Warning: Some copies of the text have a typo in #24. It should be cos3 in the force, not cos2.
3.6 # 27 (start by computing C'(w))
3.6 Iode project on resonance and transience .
Hint. To find the period of beating, theoretically, you will want to use a "sum to product" trig identity.

Due Friday 9 April in class.
Homework # 7
3.3 # Problem on solving large linear systems (the problem is in Part 2 of the document). This Problem shows you how to solve large systems of simultaneous linear equations, using Matlab.

3.5 # 6, 8, 13, 28, 34, 43, 50
and an optional no-credit problem: 59
For some of these problems, you will want to use the comment at the bottom of the Undetermined Coefficients handout about what to do if f(x) on the right side of the DE consists of a sum of more than one term.
You can check your answers with Wolfram Alpha.
For 43(a), the suggestion in the textbook can be expanded to say: cos(3x) + i sin(3x) = ei(3x) = (eix)3 = (cos(x) + i sin(x))3.
For 50, you should use the Variation of Parameters formula (33) on page 209; you do not have to go through the procedure with u1 and u2 on pages 208-209.
For optional problem 59: the DE is equidimensional, but it has equal "roots". That is why the logarithm is needed in yc.

Due Friday 19 March in class.
Homework # 6
3.4 # 2, 4 (the mass is .25 kg; note the first sentence helps you determine k: it says a force of 9 N would stretch the spring by .25 m),
5 (first read the instructions above the problem; note there is no friction in this problem),
6 (losing 2 minutes 40 seconds per day means that the pendulum takes 24 hours plus 2 minutes and 40 seconds at the equator to go through as many oscillations as it went through in 24 hours at Paris)
12abcd (note it is easier to use the metric values: g=9.8 m/s2 and R=6370 km); note I am not sure that there is a simple way to explain part d, but you might enjoy trying...
13, 14
(and for # 13, 14: determine whether the system is overdamped, critically damped, or underdamped. Check: plot your function x(t) using a graphing calculator or Wolfram Alpha, to make sure the graph looks like Figure 3.4.14 or 3.4.15.)
23 (here m=100, and you should express the frequencies w0 and w1 in radians per second, in parts (a) and (b) respectively),
31 (you can use the binomial series, which is just the Taylor series of f(x)=(1+x)a around x=0; the first two terms of the series give the approximation (1+x)a = 1 + ax + ..., which is one of the most useful in all of science - but only use it if x is small!)
3.3 # 28 (guess the roots by plotting the characteristic polynomial; then check that your guesses are correct; then write the general solution)
48 (when you get equations for A,B,C, you can notice that A=B=C=1/3 is the solution)
Due Friday 12 March in class.
Homework # 5
3.3 # 22, 43c (Hint. You can check your work using Wolfram Alpha. e.g. enter 2-2i sqrt(3) at the website)
Also do: Iode Project 3
3.4 # 19 - check your answers at the back of the book; you can make plots using Wolfram Alpha, and copy the plots (by right-clicking) into a Word document for printing
3.7 # 2,
11, but after finding Isp(t), put it into the form I0cos(wt-d) instead of the form I0sin(wt-d) that the textbook asks for. Also, find the "transient" current Itr(t), that is, the solution of the associated homogeneous DE; it is called transient because it tends to 0 as time t tends to infinity. Note you cannot evaluate the constants in this transient solution, because you do not know the initial conditions.
Due Friday 5 March in class.
Homework # 4
3.1 # 4, 2 (and also do problem 2 using y1(x)=cosh(3x), y2(x)=sinh(3x); which way of satisfying the initial conditions seems easier: using the exponential solutions, or cosh and sinh? why?),
19 (comment: this example shows that the Superposition Principle is false for nonlinear differential equations),
27 (the "associated homogeneous equation" is y''+py'+qy=0; comment: we will use this decomposition principle many times in the course, to express the general solution to the nonhomogeneous equation as the sum of the general solution yc to the homogeneous equation and a particular solution yp to the nonhomogeneous equation),
34, 36, 40 (your answers to these three problems should clearly state (i) the characteristic equation, and (ii) the solution y(x))
44, 45, 47, 48
Hint. You can check many of your answers using Wolfram Alpha.
Due Friday 26 February in class.
Homework # 3
1.5 # 38 (comment: as a more real-world example, you might consider two lakes instead of two tanks).
Just for fun (not to turn in), try 42 - it explains that hailstones fall much slower than rocks or skydivers because hailstones get heavier as they fall! You may assume the hailstone is spherical.
1.6 # 26 (the lecture notes might give you a hint about how to do this problem)
2.2 # 4 ("critical point" means "equilibrium value"),
23 - to help justify your answers to parts (a) and part (b), draw the phase lines and indicate the equilibrium values and their stability. Note. The DE is "logistic" if it has the form dx/dt=ax-bx2 where the constants a and b are positive (see page 81).
Also: (c) Interpret the meaning of the constant kM (hint: suppose x is very small, so that the term -kx2 in the DE can be neglected).
Then: (d) Interpret the meaning of the extinction criterion h > kM in part (b).
2.4 Iode Project II. (First make sure you understand Iode Lab II.)

Feedback: download the Feedback on Iode Project II.

Due Friday 12 February in class.
Special office hour in the computer lab Engineering Hall 406B1, Thursday 4:30-5:30pm. (No office hour 3:30-4:30pm that day.)
Homework # 2
1.3 Iode Project Ib
1.4 # 28, 65 (hint: write t=0 for the time of death and t=a for the time elapsed from death until noon; your task is to find the value of a),
66 (hint: how deep is the snow on the road at time t, ahead of the snowplow?),
69 (here v=dy/dx; hint: 1+sinh2(t)=cosh2(t); consult your calculus book to remind yourself about the properties of the hyperbolic trig functions sinh and cosh)
Comment: the cable in problem 69 might be an electricity transmission line suspended between pylons.

Feedback on 1.4 #66:
(a) [Modeling] Let t=0 be when the snow starts to fall. Consider the short time interval from t to t+dt. During that time, the amount (volume) of snow cleared by the plow equals c dt where the constant c is the rate of snow clearance. On the other hand, the volume of snow cleared by the plow during that time interval can also be expressed as (bt)w dx, where dx is the distance traveled by the plow, w is the width of the plow and bt is the depth of snow on the road (remember the snow falls at a constant rate, so the depth of snow on unplowed parts of the road is proportional to t; here b is our constant of proportionality). Setting equal the two expressions for the amount of snow cleared, we find c dt = btw dx, which can be rewritten as the Differential Equation k dx/dt = 1/t with the constant k = bw/c.
Comment. Notice we used a general principle of modeling: consider a small dt or dx, and describe "what happens" during that short interval. If you can find two different expressions for the same thing, then set the two expression equal.
(b) [Solving] The DE can be written dt/dx=kt, so the solution is t=Cekx. Let t=a be the time 7:00am. The problem tells you the following three pieces of information: t(0)=a, t(2)=a+1, t(4)=a+3. This gives three equations for the three unknowns a,C,k. Use the equations to show that a=1, so that 7:00am is one hour after the snow started. Conclude it started snowing at 6:00am.

Due Friday 5 February in class.
You must use a cover sheet (see above), and the homework must be stapled together.
Homework # 1
1.1 # 16, 36 (and explain your answer with a brief sentence),
43a (use the separable method to solve the differential equation),
44 (sketch solutions for part a and for part b), 46 (and sketch the solution)
1.2 # 35 (comment: thus in order to double the impact velocity, you must quadruple the initial height), 40
1.3 Do Iode Project Ia
1.4 # 48 (see page 40 for an explanation of "half-life")

Feedback on 1.4 #48:
Use the given halflife information to determine the decay constants k1 and k2 for U-235 and U-238 respectively. Then by page 37 you have the formulas N1(t)=C1 exp(-k1t) and N2(t)=C2 exp(-k2t) for the amounts of U-235 and U-238. The problem says to assume there was the same amount of each when t=0, so C1 = C2. And the problem says that at the present time t=T we have 137.7N1(T) = N2(T). (Common error: putting 137.7 on the other side of the equation. Check your formula by re-reading the problem - it says that there is a lot more U-238 than U-235, so N2(T) must be much bigger than N1(T).) Solve the last equation for T, to get the age of the universe.

Due Friday 29 January in class.
You must use a cover sheet (see above), and the homework must be stapled together.

[Answers to many problems are in the back of the textbook.
The Student Solutions Manual has worked solutions for most odd-numbered problems.]