Math 2a - Practical Track

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Differential Equations

Fall 2011-12



MWF 10:00 AM // 22 Gates


Course Description | Policies | Textbooks | Lecture Notes | Handouts | Homework | Sections


Instructor: Dinakar Ramakrishnan, 278 Sloan, 626-395-4348,


Lead TA: Daiqi Linghu, 382 Sloan,

Course SecretaryKristy Aubry, 253 Sloan, 626-395-4087,


Feedback Form





 FINAL EXAMS can be picked up in the Math Department office (253 Sloan) Thursday, December 1 after 1pm and until 5pm. A limited amount of copies will be available near the homework boxes after hours.
A final review will be held by Kevin Linghu, in 151@Sloan, 5:00-6:00pm on Friday, Dec 2nd.
D. Ramakrishnan will briefly review in class  the salient points of what was done after the midterm on Friday,   Dec. 2, 2011.

All the Lecture Notes on the material needed for the Final Exam are now posted (see below).





Section 1 - Daodi Lu
Office Hours: Saturday, 3 pm - 4pm
155 Sloan, 626-395-4081


9:00 AM, Thurs.  

151 Sloan


Section 3 - Xiang Ni
Office Hours: Sundays, 3 pm - 4 pm
155 Sloan, 626-395-4081


10:00 AM, Thurs. 

151 Sloan


Section 4 -Cai Li Shen
Office Hours: Mondays, 8 pm - 9 pm
155 Sloan, 626-395-4081

10:00 AM, Thurs.

153 Sloan


Section 5 - Brian Hwang
Office Hours: Thursdays, 2 pm - 3 pm
155 Sloan, 626-395-4081

1:00 PM, Thurs.

102 Steele


Section 6 - Gaurav Sinha
Office Hours: Fridays, 6 pm - 7 pm
155 Sloan, 626-395-4081

2:00 PM, Thurs.

142 Keck


Section 7 - Daiqi Linghu*
Office Hours: Mondays, 3 pm - 4 pm
153 Sloan, 626-395-4366

2:00 PM, Thurs.

102 Steele

Section 8 - Liling Gu
Office Hours: Mondays, 7 pm - 8 pm
 356 Sloan, 626-395-4354

1:00 PM

125 Baxter






Homework: Homework will be posted online a week in advance and is due on the following Tuesday by 12pm (noon) in the appropriate box outside room 253 Sloan.

Grading: 50% homework, 25% midterm, 25% final. The midterm and final are 3-hour take-home exams that will be handed out in class.

Collaboration and Tools: Collaboration is allowed on the homework, but your write-up must be in your own words and may not be copied. Collaboration is not allowed on the exams. For homework sets you may use books and notes, but you may not consult homework sets or solutions from previous years of Math 2a Practical. For exams you may use your returned homework sets, their solutions as posted on this webpage, and the class textbook. For exams you may also use notes that you took in lecture. What if you miss a lecture? For no more than 2 lectures before the midterm, and for no more than 2 lectures between the midterm and final, you may use notes that you have copied by hand from a classmate. Computer/calculator usage: for homework sets and exams, unless explicitly stated otherwise, you may use a computer/calculator to plot direction fields. Use these plots only as guides, as you will not receive credit for justifications of the form "I know this because my computer/calculator told me so".

Late Policy: Each student will be allowed one homework extension, so long as they email their TA by midnight the night before it is due, asking for an extension and providing a reason. Such extensions will be until Wednesday at 12pm. Late homework sets without extensions (or those turned in after an extension deadline) will be graded as follows: if turned in less than a week after the original deadline (turned in before the following Monday, 12pm), the score will be multiplied by 50%. If turned in more than a week after the original deadline it will receive no credit. Under exceptional circumstances, a note from the Dean or Health Center may be considered. Late exams will not be accepted except under exceptional circumstances.

: Go to the recitations! Each week, a portion of the recitation will be dedicated to introducing some computational (or even theoretical) aspect which the Instructor will not have time to present in class. In general, get to know your TA and bug him/her to death (figuratively speaking) with your questions, and try to fill in all the holes in your understanding of the subject.






  • Elementary Differential Equations and Boundary Value Problems 9th Edition by Boyce & DiPrima, ISBN  0470383348


Material Covered





Week 1
Introduction, Direction fields, classification, integrating factors, separable equations. modeling with first order equations, linear versus non-linear ODE's - Secs. 1.1-1.3, 2.1-2.4 in the text
Week 2
Exact equations, Integrating factors, Difference equations, and (if time permits) population dynamics -
Secs. 2.6 – 2.9, and (if there is time) 2.5

Week 3
More on Difference Equations, Euler’s method, Existence and Uniqueness for first order ODE’s – Sections 2.7 – 2.9 in the text

Week 4

Linear systems of fist order linear homogeneous equations
- sections 7.1, 7.4-7.7 of the text
(The students should review on their own the material from Ma 1b on matrix equations and eigenvalues (including complex ones) - roughly sections 7.2,
7.3 of the text.)                                                                                       

Week 5
More on linear systems, especially repeated eigenvalues, and the exponential of matrices – Sections 7.7 – 7.8 in the text; plus midterm review

Week 6
More on repeated eigenvalues and exponentials, inhomogeneous linear systems, 7.9, 8.1 in the text
Week 7
Linear ODE’s of higher order, making use of results from linear systems of first order; vibrations, damping, resonance; variation of parameters, undetermined coefficients – Selections from chapters 3, 4 in the text, but with a different slant
Week  8
Series solutions (for second order linear ODE's), near ordinary points and regular singular points, Bessel's equation - Chaper 5 of the text
Week 9 The Laplace Transform, Discontinuous forcing, Convolution - Chapter 6 of the text
Week  10
Truncation Errors, The Runge-Kutta method    –  Sections 8.1-8.3 in the text


Lecture Notes





Oct. 17, 2011
Notes on complex numbers, functions, and factoring
Oct. 24, 2011 Notes for Lectures 1-5
Oct. 29, 2011 Notes for Lectures 6-9
 Nov. 9, 2011
 Notes for Lectures 10 -12
 Nov. 14, 2011  Notes for Lectures 13-15
 Nov. 16, 2011  Notes for Lectures 16-18
 Nov. 21, 2011  Notes for Lectures 19-21
 Nov. 24, 2011  Notes for Lectures 22-23
 Nov. 28, 2011  Notes for Lecture 24
 Nov. 30, 2011  Notes for Lectures 25-26




Due Date



October 4 @ 12 PM
Homework 1
October 11 @ 12 PM
Homework 2

October 18 @ 12 PM
Homework 3

October 25 @ 12 PM
Homework 4 revised
November 8 @ 12 PM
Homework 5 revised

November 15 @ 12 PM
Homework 6

November 22 @ 12 PM
Homework 7

December 1 @ 4 PM
Homework 8



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