The weekly homework session is Monday 5-6pm in Altgeld 141, and
office hours are Tuesday 3-4pm and Thursday 3-4pm in Altgeld 376.
Let me know by email if you want
to meet at some other time.
The median scores on the Homeworks so far are:
9.5/10, 9.5/10, 14/17, 8.75/10, 9.5/10, 8.5/10, 14.5/15, 8.5/10, 8/10,
10.5/10, 10.5/12, 10.5/10
The median scores on the Quizzes so far are:
1, 1, .75, .625,
(all out of 5)
The top score on Test 1 was: 71/75, median 55/75.
Approximate A: 60-75, B: 50-59, C: 40-49, D: 30-39, F: 0-29.
The top score on Test 2 was: 85/75, median 50/75
Homework procedures
Homework can be turned in during class, or to the tray outside
my office 376 Altgeld Hall. All homework must be stapled.
I encourage you to work in groups of two or three on the homework
problems. But each person must write up his/her own
solution individually, to turn in.
Late homework will be accepted only if you have made prior arrangement
with me.
I will drop your 2 lowest homework scores, over the semester, except that
some of the assignments will be announced as mandatory (i.e. they
cannot be dropped).
| Homework # 1 |
Hand in the following: Homework 1 Supplement (print this off) Section 1.1: 9, 13, 19, 24, 33 (and solve for v(t)), 35 Section 1.2: 3, 10, 18 (acceleration is nonconstant, so eq. (11) does not apply) |
Due Tuesday 23 January, 5pm |
| Homework # 2 |
Section 1.3: Project 1 Section 1.4: 16, 17, 28, 47 |
Due Tuesday 30 January, 5pm |
| Homework # 3 |
Section 1.5: 14, 19, 20, 29, 38 (In #29, first find the derivative of erf(x), before solving the problem. Comment on #38: cascades arise naturally when studying the Great Lakes, or water purification systems.) Section 1.6: 16, 14, 23, 61 (Aside: #61 is the default example in Iode.) Section 2.2: Consider the following two equations: (a) (dx/dt) = x^2 - x - 6 (b) (dx/dt) = cos2x For each equation, (i) find all "equilibrium" x-values (also called "critical points", on p. 91), (ii) draw the phase line (vertically!), (iii) determine the stability or instability of all equilibrium x-values, (iv) plot enough solution curves (either by hand or with Iode) to make the picture clear in all regions of the tx-plane, (v) find the general solution by hand (for example by using the method of partial fractions to integrate, or by looking up integrals from a table). [Check: do your plots in part (iv) basically agree with your solution formula in part (v)?] Section 2.4: Project 2 (*Question 5 on this project is for extra credit.) |
Due Tuesday 6 February, 5pm
Mandatory (may not be dropped); worth 1.7 times a normal HW |
| Homework # 4 |
Section 3.1: 5, 12, 19, 27, 36, 37, 48 Section 3.2: 6, 9, 25, 26 (in problems 6 and 9, use the definition of linear dependence and linear independence from class, as on the Section 3.2 handout; thus do NOT use Wronskians) Section 3.3: 8, 22, 41, 43(b) (but only do -1+i 31/2). Note that in #22, the solution tends to zero as x tends to infinity, but this is not the case in #8. Illustrate these observations by creating a plot for each problem (clearly labeling each plot, and choosing the domain and range carefully to show the relevant features of the graphs). You can use the "2nd order linear" module of Iode to create the plots, if you wish. |
Due Thursday 15
February, 5pm |
| Homework # 5 |
Section 3.4: 3 (write 20cm as 0.2m), 5 (first read the instructions above the problem, and read pages 183-184), 6 (this means the pendulum takes 24 hours 2 minutes and 40 seconds at the equator to complete as many cycles as it does during 24 hours at Paris), 13, 14. (And for # 13, 14: determine whether the system is overdamped, critically damped, or underdamped.) Section 3.4: 23 (here m=100, and you should express omega in radians per second), 31 (you can use the binomial series on page 498; it is just the Taylor series of f(x)=(1+x)alpha around x=0) Section 3.5: 3, 4, 6 |
Due Wednesday 28 February, 5:00pm |
| Homework # 6 |
Section 3.5: 10, 22, 26, 29, 34, 43, 50 (use the variation of
parameters formula; incidentally, for fun you can do 59 by variation of
parameters, where the homogeneous DE is
equidimensional but has equal roots and hence yc needs a
logarithm) Section 3.6: 5 (undamped and forced not at natural frequency - beating!), 6 (undamped and forced at natural frequency - resonance!), 11 (damped and forced), 18 (damped and forced), 24 (undamped and forced at two frequencies, hint: cos3(u)=(3/4)cos(u)+(1/4)cos(3u) by Section 3.5), 25 (damped and forced) |
Due Wednesday 7 March, 5:00pm |
| Homework # 7 |
Section 3.6: 27 (hint: it is easier
to work with 1/C(w)2 and then draw conclusions about C(w); note that in part (b), you are trying to show C(w) has
graph shaped roughly like in Figure 3.6.9)
Section 3.6: Iode Project III on Forced Oscillations Section 4.1: 5, 24 Section 5.1: do 1-6 but do not turn in (check the answers in the back of the book); turn in: 12, 20, 22, 27 (but on 22 and 27, don't use Wronskians; instead I want you to check linear independence of the solution vectors at t=0 geometrically, by verifying that no one of the vectors can be written as a linear combination of the others; the key point for linear independence on 27 is that if v1 and v2 are vectors then the linear combination c1v1+c2v2 is a vector lying in the plane spanned by v1 and v2), 31, 36. |
Due Wednesday 14 March, 5:00pm
Mandatory (cannot be dropped) Worth 1.5 times a normal HW. |
here. (If you have
comments specifically about Iode or the computer projects, then they are
best submitted
here).
| Homework # 8 |
Problem A. Consider the second order, constant coefficient homogeneous
equation x''+px'+qx=0.
(i) Find the relation between the roots r1 and r2 of the characteristic equation and the eigenvalues lambda1 and lambda2 of the associated 2x2 system of equations for the dependent variables x and y=x'. (ii) Deduce from (i) what kind of phase portrait must be associated with an underdamped oscillator, and with an overdamped oscillator. (Consult your class notes for the names of the types of phase portrait we have encountered so far e.g. center, sink.) Section 5.2: 4, 8, 11 (and for 4, 8, 11, also name the type of the phase portrait), 20 (remember the cofactor formula (20a) on page 286 for computing determinants that are 3x3 or bigger; usually we expand across row i=1), 37 (and explain why you do not see oscillations, when you graph x1(t), x2(t), x3(t)), 49 (use Matlab commands from page 314 to find the eigenvalues and eigenvectors, because this system is too big to do comfortably by hand) Section 5.3: 3, 9, 12, 14 (and explain intuitively how this is possible - hint: plot F(t), x2(t)) Problem B: Analyze Section 5.3 #3 with the addition of forcing [0 cos(wt)]T (that is, force the second component at frequency w), by following the model of Example 3 on page 323. (i) In particular, calculate the amplitude response, and then plot it like in Figure 5.3.10. |
Due Wednesday 28 March, 5:00pm |
| Homework # 9 |
Section 5.4: 7, 11 Section 5.5: 9, 19, 23, 27, 32 Section 5.6: (follow your class notes: for first order constant coefficient nonhomogeneous problems you can use the integrating factor method or the eigenvector decomposition method; for first order variable coefficient nonhomogeneous problems you can use the variation of parameters formula; for second order constant coefficient nonhomogeneous problems you can use the eigenvector decomposition method. I recommend not using Undetermined Coefficients for systems of equations, and especially not for second order problems, because there can be difficulties in the resonant case.) 2, 4, 14, 16, 25 Problem B continued: (ii) For the cases where the forcing frequency w equals a natural frequency, you should confirm that resonance occurs by finding a particular solution (hint: use the eigenvector decomposition method from Section 5.6 class notes) and checking that the amplitude grows to infinity as t -> infinity. Aside: notice we have two ways of detecting resonance: first by plotting the amplitude response curve and seeing at which forcing frequencies it has an infinite singularity, and second by finding the particular solution for each possible value of w and looking for which particular solutions have an amplitude that grows with t. |
Due Wednesday 11 April, 4:30pm |
| Homework # 10 | Download Homework 10. And here's the solution to a related Neumann exercise from class. | Due Wednesday 18 April, 5:00pm |
| Homework # 11 |
Section 9.3: 9, 13, 15 [Hint: for 13 and 15 you might want to use results from Example 1 in Section 9.3] Do the Fourier series project. Section 9.4 #3, 8, 9, 10. For 8,9, you are not required to explicitly find a particular solution. For 10, I want you to evaluate a particular solution. For Sections 9.3 and 9.4, refer to the handout given in class. |
Due Wednesday 25 April, 5:00pm Mandatory |
| Homework # 12 | Download Homework 12 | Due Wednesday 2 May, 5:00pm. |