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

• Week 1 (Jan. 22): No recitation due to the snow. Please read the Review Slides for Calculus Formulas.

• Week 2 (Jan. 29): Surveillance test. Slides for Recitation 1. Challenge Problem 1.

• Week 3 (Feb. 5): No recitation due to the snow. Please read the Slides for Recitation 2 (that never happened).

• Week 4 (Feb. 12): Quiz 1. Slides for Recitation 3. Challenge Problem 2.

• Week 5 (Feb. 19): Quiz 2. Slides for Recitation 4. Challenge Problem 3.

• Week 6 (Feb. 26): Quiz 3. Slides for Recitation 5. Practice Problem for 2.6. Solution.

• Week 7 (Mar. 5) : No Quiz due to the midterm next day. Please read the Review Slides and also the Review Questions. Note that example problems I have done on the blackboard are not included in the slides.

• Week 8 (Mar. 12): Quiz 4. Challenge Problem 4
  It is a shame that I did not talk about the the more convenient way of doint the computation. Please watch first MIT Lecture 13 and MIT Lecture 14. Then please read Recitation 7a: Computations using Exponential Shift Law.
  In case you are still not confident dealing with Reduction of order, please read Recitation 7b: More about Reduction of Orders. You will find the answer to the Challenge Problem 4 in this set of slides.
  In case you want to get acquainted of dealing with complex numbers, please watch MIT Lecture 6

• Week 9 (Mar. 19): Spring Break. No recitation. Please read Recitation 7a: Computations using Exponential Shift Law

• Week 10 (Mar. 26): Quiz 5 (Take-Home). Slides for Recitation 8. Challenge Problem 5
  Note: Materials for this week is not in the syllabus. You don't have to learn it if you don't like it.

• Week 11 (Apr. 2): Quiz 6. Slides for Recitation 9
  For performing Laplace transforms, the most vital step lies on partial fraction expansion. Here are some materials for practise:
  Basics about partial fraction expansion
  Cover-up method (written by Dr. Mattuck). Notice that this method does not always work.
  Summary of Practial Fraction Expansion. Here the document uses complexification to deal with quadratic factors.

• Week 12 (Apr. 9): Quiz 7. No more slides from this week on. Instead, please watch MIT Lecture 22

• Week 13 (Apr. 16): No quiz due to the midterm the next day. Please read the Review Questions for Midterm 2 to prepare for the midterm (Watch out for typos). Especially, please do Problem 9 in Section 1 and find the answers in Section 5. Also please read Section 3 and Section 4.

• Week 14 (Apr. 23): Quiz 8. Also please watch MIT Lecture 25 and MIT Lecture 26.
  Although the topic of repeated eigenvalue is not covered in class, without it the story of 2x2 linear system is incomplete. So anyway please watch the video.
  For more example problems, please read Dr. Z's notes for Lecture 19 and Lecture 20 and try Attendence Quiz 19 and Attendence Quiz 20.
  For repeated eigenvalue case, please also read Dr. Z's note for Lecture 21 and try Attendence Quiz 21.

• Week 15 (Apr. 30): Quiz 9. Also, please watch the following videos for getting acquainted to series solutions:
  http://www.youtube.com/watch?v=RJJKq7Uc-9I
  http://www.youtube.com/watch?v=oY0ItxI9xTk
  Also please review calc 2 for convergence issues.

• Week 16: No more official recitations. I'll be in my office almost every day. Please feel free to drop by. Also here is some materials that might be useful for review:
  Notes on Dr. Virbhadra's Review Session Thank anonymous student for sharing.
  Review Questions for stuffs after Midterm 2

• 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

• Please make sure you know how to differentiate and integrate and you can play with the elementary functions. You can review the issue through the slides I wrote
  Brief review of basic formulas (out of date! please find the revised version above)
  Also you can consult the famous Russian book
  3193 Problems in Mathematical Analysis
  I have the full solution to ALL the exercise problems. If you are in trouble with ANYTHING please come to me for help.

• Please find the basic rules: http://www20.csueastbay.edu/library/scaa/files/pdf/Alg%20rules.pdf
  It is required that you know these rules well. ANYONE making ANY mistakes in quizzes on these rules WILL SUFFER PENALTY!

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.

Fei Qi
Room 624, Hill Center
Department of Mathematics
Rutgers University
110 Frelinghuysen Road
Piscataway, NJ USA 08854