01:640:244 Differential Equation for Physics and Engineering, Sections 8 - 10, Spring 2014

Please find Dr. Virbhadra's website for the course here
http://math.rutgers.edu/~shwetket/Math244.html

Also please find the homework assignments here
http://math.rutgers.edu/~shwetket/Math244/HW.html

In case you have doubts and any point that is not clear, please try the MIT video lectures here:
http://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/

All announcements are posted on sakai only. Please make sure that you receive emails from the system.
Although everything is uploaded on sakai, in order to make it easier to retrive, I made some of the course materials available here.

From Week 9 on, I have been so busy that I did not manage to keep this website updated. Now that it is summer and I have to make a new page for the new course. Let me put the rest of materials here.

• In case you messed up with integration by parts in the midterm, here are two examples and twenty-five exercises (1211 - 1235) for you to practice. The answers are available here.

• In case you are still not very familiar with any kinds of substitutions, here are five examples and twenty exercises (1191 - 1210) for you to practice. The answers are available here

• In case you don't know how to draw a graph a function, please find instructions here together with two very explicit examples. You can certainly practice any of those seventy-seven exercises (916 - 992) but you probably need to look for answers in mathematica / maple.

• Also when you integrate 1/x, you don't have to care about the absolute values ONLY in the scenarios I mentioned in the slides of Recitation 3, Page 48. For cases other than that, you SHOULD care about the absolute values.

• The arbitrary constant that is produced by integration is NOT NECESSARILY A PURE CONSTANT. For first order linear ODE, the constant ALWAYS appear with the inverse of integration factor!

• I really don't enjoy talking about this here, but THE SQUARE ROOT OF a+b IS NOT THE SQUARE ROOT OF a PLUS SQUARE ROOT OF b. Also, 1 OVER a+b IS NOT 1 OVER a PLUS 1 OVER b. In quizzes if you make such mistakes, don't you ever ask for mercy.

• When you find the integrating factor, (M_y-N_x)/N DOES NOT STAND FOR u(x) BUT FOR u'(x)/u(x). So to get this factor, you have to INTEGRATE!!

• When you find the function F(x,y) for an exact solution, the problem is not over. You are asked the solution of the ODE, which is expressed as F(x,y)=C. F(x,y) itself is a function instead of a solution to an ODE.

• Many people messed up with the factorization. Here is what you should do

r^4 + 2 r^2 + 1 = (r^2 + 1)^2
s^2 - s - 6 = (s - 3)(s + 2)

Also for the inverse Laplace, the polynomial s^2 - 4s + 5 CANNOT be factorized unless you get complex number involved. To gets its ILT, one should complete the square:

s^2 - 4s + 5 = (s - 2)^2 + 1

then use exponential shift formula and the table.

• Also for the problem 1a, the first try actually succeeds: what you have in the complementary solution is cos t and sin t. And your first try template is A cos 2t + B sin 2t, which does not coincide to the complementary solution! Please read the review sheet section 3 really carefully.

• For problem 1b, the template for the constant summand 4 actually fails twice. So you should set up Et^2 at the end. Also the template for the summand te^t should be (Ct + D)e^t. When you have a polynomial in your template you should finish it all the way down.  Please read the review sheet, do Problem 9 in Section 1, and check your answers in Section 5. Also please read Section 3.

• For either LT or ILT, THERE IS NO PRODUCT FORMULA at least to what you have known. L(fg) is not L(f)L(g) and L^-1 (FG) is the convolution product of L^-1(F) and L^-1(G) (See Section 6.6 in the book) which is not L^-1(F) L^-1(G)  You should always stick to the table and don't use anything beyond that!

• For problem 2a, many people messed up with the signs when dealing with the term L(-y'), which is - s L(y) + y(0) = - s L(y) + 1. Although this will lead to a disaster in the steps following, I only deducted 2 points if all the rest steps are correct, even though your solution is far from the correct one.

• For problem 2b, although many people get full credit, very few people actually got the right answer. Remember you are looking for a linear system of ODE, the standard form of which looks like

x_1' = f_1(x_1, ..., x_n)
x_2' = f_2(x_1, ..., x_n)
.........
x_n' = f_n(x_1, ..., x_n)

so the expected answer is

x_1' = x_2
x_2' = x_3
x_3' = x_4
x_4' = x_1

In grading, I gave full credit if I found all expressions above somewhere in your solution. But you should write them together. Remember this in the final.

For 244 students, I have two requirements

If you have difficulties in these algebra issues, a series of link is provided for help.

• If you don't know how to manipulate logarithm, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch10_SE.pdf
Please read Section 10.5 on page 45 in the pdf file (page 733 in the book), try all example problems, and do Exercise 44 - 61 on page 51 in the pdf file (Page 740 in the book).

• If you are not very fluent with the quadratic equations (e.g. always use the root formula), please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch08_SE.pdf
Read Section 8.1, 8.2, try all example problems, and do Exercise 66 - 83 on page 23 in the pdf file (Page 573 in the book). Make sure you understand all the related methods

In particular, if you have never seen criss-cross factorization before, please check the youtube videos
Criss-Cross Method 1, Criss-Cross Method 2, Criss-Cross Method 3 and Criss-Cross Method 4.

• If you have never seen matrices before, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch03_SE.pdf
Read Section 3.6, try all example problems, and do Exercise 15 - 23, 46 - 49 on page 51 - 52 in the pdf file (page 227 - 228 in the book).
Read Section 3.7, try all example problems, and do Exercise 2 - 7, 20 - 25, 35 - 40 on page 63 - 64 in the pdf file (page 239 - 240 in the book).
After you work on this topic, try the problems of the attendence quiz at Lecture 15 and you will find it easy to play.

• If you keep on making mistakes on exponentials, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch01_SE.pdf
Read Section 1.8, try all example problems, and do Exercise 59 - 84 on page 88 in the pdf file (page 88 in the book).

• If you don't know how to divide a polynomial, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch05_SE.pdf
Read Section 5.3, try all example problems, and do Exercise 27 - 42 on page 31 in the pdf file (page 339 in the book).
After you have done the work, please compare to the technique I used on dealing with t/(t+1) or -2-t/(t+1) in class. You will see that this is actually the simplest example of division.

• If you are not fluent on simplifications of rational functions, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch06_SE.pdf
Read Section 6.1 - 6.4, try all example problems, and do Exercise 29 - 48 on page 61 - 62 in the pdf file (page 463 - 464 in the book).

• If you are not fluent on playing with trigonometric functions, please find
http://www.eht.k12.nj.us/~staffoch/Textbook/chapter04.pdf
Read Section 4.3, make sure you memorize the table of the values of sine, cosine and tangent on usual special angles on page 23 of the PDF file (page 279 in the book)
and do Exercise 17 - 26 on page 28 of the pdf file (page 284 in the book)
Read Section 4.5, make sure you can recognize, distinguish different graphs of the trignometric functions and manipulate them by scaling and translation, and do Exercise 3 - 14, 23 - 16 on page 48 in the pdf file (page 304 in the book)

• If you are not fluent on factorizing polynomials, please find
http://people.ucsc.edu/~miglior/chapter%20pdf/Ch05_SE.pdf
Read Section 5.4, try all example problems and do Exercise 51 - 70 on page 40 of the pdf file (page 348 of the book) .
Read Section 5.5, try all example problems and do Exercise 9 - 46 on page 52 of the pdf file (page 360 of the book).
Read Section 5.6, try all example problems and do Exercise 43 - 70 on page 61 of the pdf file (page 369 of the book).
Read Section 5.7, try all example problems and do Exercise 1 - 66 on page 67 of the pdf file (page 375 of the book).
If you really do all the exercises, then after that you may probably find yourself addicted to playing such a game. I don't recommend to resist such an addiction. Just do all other exercises and it will accelerate your speed greatly in solving problems in Lecture 8 - 16.

Notice: The books seem to be developing and page numbering might change. But the section number together with the exercise numbering will be invariant. If you did not find the exercise and the reading materials at the pages I told you, just scroll around, or use Ctrl+F to find sections.

Fei Qi
Room 624, Hill Center
Department of Mathematics
Rutgers University