Practice Problems for ODE final

List of problems recommended by Charu!

Chapter 3

3.1:  Example 1, Example 2

Problem Set 3.1:  Q7 – Q11

Problem Set 3.3:  2, 3, 5, 6, 10, 13, 15

3.6:  Example 1, Example 2

Problem Set 3.6: 2, 3, 5, 7, 8, 9, 10, 12, 13, 14, 15

Chapter 5

Table 5.1

Problem Set 5.1:  1, 2, 3, 4, 5, 6, 9 , 12, 14, 16, 17, 19, 20, 22, 24, 26, 27, 28, 32, 33, 34, 36,, 37, 38, 39.

5.2: Example 1, Example 2, Example 3, Example 4, Example 7, Example 8

Problem Set 5.2:  1, 2, 4, 7, 9, 13 – 20

5.3: Example 1, Example 2

Problem Set 5.3:  2, 4, 6, 7, 8 – 13, 21, 22 – 30.

5.4: Example 2, Example 3

Problem Set 5.4:  1-16

5.5: Example 1, Example 2, Example 3, Example 4

Problem Set 5.5:  1 – 33

Note:

You can use formulas listed in table 5.1 without proof. Apart from this table, other formulas for Laplace and Inverse Laplace of the function given by various theorems in Chapter 5 can also be used without proof. Any other formula used must be accompanied by a brief reasoning.

ODE Final Exam

The Final Exam for ODE will be based on the following sections of the 8th edition of Kreyszig's Advanced Engineering Mathematics:

Ch 1: All, except sections 1.2 and 1.9.
Ch 2: All, except sections 2.4 and 2.13--2.15.
Ch 3: 3.0 to 3.3. From 3.6, only method of undetermined coefficients when RHS has no term which is a solution of the homogeneous equation.
Ch 5: 5.1 to 5.5.

The marks distribution will be Ch 1 (20), Ch 2 (20), Ch 3 (20), Ch 5 (40).

Further information may follow.

Mathematics Seminar:
Speaker:     Nipun Thakurele 
Title:        Anticipating the Stock Prices
Schedule:   December 14, 2011. Period 5 (13:20-14:10)  
Venue:        Class Room for Section C

Penrose Tiling video

As a follow up to the impromptu description of Penrose tiles by Prof. Fozia Qazi in our Thursday class, here is an animation of their properties:

Part 1 - Symmetries



Part 2 - Scaling



The creators of these videos, Maurizio Paolini and Alessandro Musesti, teach Mathematics at the Department of Mathematics and Physics Niccolò Tartaglia at the Catholic University of the Sacred Heartk in Brescia, Italy. The website for this project is frecceaquiloni.dmf.unicatt.it/

A couple of puzzles from Russia

A generation (or two) ago, the most interesting textbooks in India came from the USSR - they were typically written by the best mathematicians and scientists of that country, and mixed a sense of fun into the excitement the authors obviously felt about their subjects. Some of these books are again becoming available, but through foreign publishers and not at the old prices of a few rupees per book! One that has been reborn is a slim volume titled "Functions and Graphs" by I.M. Gelfand and two others. Here are two puzzles from this book:

1. An honest merchant knew one arm of the weighing scales he was using was slightly longer than the other. In an effort to be fair, he decided to weigh half the merchandise to each buyer on one pan, and half on the other. Did he gain or lose as a result?
2. Seven matchboxes are arranged in a circle. The first contains 19 matches, the second 9, and then 26, 8, 18, 11, and 14 respectively. Matches can be moved from any box to an adjacent box. How should they be shifted so that all the boxes have an equal number and we have shifted as few matches as possible?

Precalculus- Notes on Cardinality 2

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Cardinality_2

Precalculus - Notes on Cardinality 1

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Cardinality_1

Precalculus Assignment 2

SNU-Precalculus-2011-Assgt2

Move on Dude

ODE - Nov 4 Exam - Practice Problems

From Kreyszig, 8th Edition:

Problem Set 2.2 – Q13, Q17, Q22

Problem Set 2.3 – Q8, Q16, Q20

Problem Set 2.6 – Q4, Q6, Q14, Q16

Problem Set 2.7 – Q4, Q6, Q8, Q10, Q12

Problem Set 2.8 – Q4, Q6, Q8

Problem Set 2.9 – Q13, Q15, Q17, Q21, Q22

Problem Set 2.10 – Q1, Q3, Q5, Q11, Q13

Problem Set 2.11 – Q1, Q7, Q13

Problem Set 2.12 – Q5, Q7, Q11, Q13, Q14

Maths on the Web 1

We have just added a "Fun Links" box on the right. The first set of links come from Nishant Suri. He explains his selections:

"Timothy Gowers is a British mathematician who is most well known for work in analysis and combinatorics. He was awarded the Fields Medal in 1998. Another feather in his cap is editorship of the excellent Princeton Companion to Mathematics. Apart from his research, Gowers is known for his very popular blog, which is among the most well written math blogs I've come across. His articles are often insightful, and almost always exemplary in their style of exposition. The fact that he is also a prolific writer is an added bonus, and means that one is delighted with new articles more often than on most other blogs. His recent series of articles on basic logic might be of particular interest to the B.S. students here.

While on the subject, let me also recommend this wonderful set of multiple choice quizzes on topics ranging from basic logic and functions to symmetries of the Fourier transform. Again, the B.S. students might benefit from some of these quizzes, especially the one on logic and the one one functions. Most of these quizes have been designed by Terrence Tao, who is an Australian mathematician known for his prodigious talent and remarkable breadth of mathematical work, which includes topics in harmonic analysis, partial differential equations, combinatorics, analytic number theory and representation theory. He was awarded the Fields Medal in 2006. He is also an avid blogger, and his blog articles include technical notes on various topics in analysis.

Mathematics now is more of a collaborative discipline than it perhaps ever was. The internet has revolutionized how people interact with each other, and it has brought the marketplace of ideas into almost every living room. When it comes to mathematics research, MathOverflow is a great place to ask for (and offer) technical help. (Everybody needs help sometimes, right?) It is not a discussion forum, nor is it a resource for home work problems. It is, instead, a place where people can ask specific and well formed questions regarding their mathematics research, and get specific and usually very reliable answers. It is one of the most effectively moderated Q & A fora around.

If you're looking for help with more mundane university mathematics, including help with homework problems, fret not! Math Stack Exchange is your best bet in such situations.

+Plus Magazine is an online magazine published by the Millennium Mathematics Project at the University of Cambridge. It is fun to read and includes feature articles which describe applications of mathematics to real world problems, a news section and mathematical puzzles and games.

Wolfram Alpha is a novel approach to handling natural language queries by sifting through heaps of data and coming up with answers, instead of results. It is thus an "answer engine" rather than the now ubiquitous "search engine". With Mathematica running in the background, it is well suited to answer high-school level mathematical questions. For example, "lim(x->0) x/sin x" yields the expected result, 1, as well as a possible derivation using L'Hospital's rule, a plot, and the series expansion. If you've never played around with it, I suggest you do! "

Precalculus Notes - Logic 2

Logic 2

Diwali Greetings!

(This "fractal flame" was composed using the software Apophysis)

Precalculus Notes - Logic I

Logic 1

Precalculus Notes 3 - Images & Preimages of Sets

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Functions 3

Dance Moves

From Facebook:

ODE Assignment 1 Solutions

Solutions to some of the ODE first assignment problems. Scroll through the embedded document using the controls at its base.

Assignment 1

Student seminar this week.

Title:        Variants of Axiom of Choice - IV
Speaker:   Nishant Suri
Schedule: Wednesday, September 28. Period 5 (13:20 -14:10)
Venue:      Tutorial Room3

Precalculus Notes - Functions and their inverses

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Functions 2

Syllabus for ODE 1st midterm

The first midterm for the ODE course will cover the first chapter of Advanced Engineering Mathematics by Erwin Kreyszig (8th Edition). However, Sections 1.2 and 1.9 are excluded.

An interesting discussion on "Proofs without words"!

Can you give examples of proofs without words? In particular, can you give examples of proofs without words for non-trivial results?
As it turns out, many people have a lot to say. Follow the link:
http://mathoverflow.net/questions/8846/proofs-without-words

Is it an appetizer or a main course delicacy!

The following question evolved during a lecture in the Linear Algebra course for M.S. students:
Do we have an explicit basis for the vector space V:= {($x_1, x_2, \ldots, x_n, \ldots ): x_i \in \mathbb{R}, \forall i \geq 1$} over the field $\mathbb R$ of reals? Or, has it been proved in literature that one cannot get hold of an explicit basis for this vector space?

Precalculus Notes - Introduction to functions

First set of notes on Functions - from the Precalculus course for B.S. Mathematics.

Functions - Part 1

Student Seminar this week!

Title:                Variants of Axiom of Choice - II

Speaker:           Nishant Suri and/or Manisha Jain 
Day and Date:  Wednesday, September 14, 2011 

Time:                Period 5, 13:20-14:10
Venue:              Tutorial Room 3

Student Seminar

Title:               Variants of Axiom of Choice 

Speaker:         Nishant Suri 
Day and Date: Wednesday, September 7, 2011 

Time:               Period 5, 13:20-14:10
Venue:             Workshop

ODE - Assignment 1

Due Date: September 9 September 12

1. A tank contains 800 gal of water in which 200 lb of salt is dissolved. Two gallons of fresh water runs in per minute and 2 gal of the mixture in the tank, kept uniform by stirring, runs out per minute.
   
   1. Form a differential equation for the amount of salt in the tank as a function of time.
   2. How much salt is left in the tank after 5 hours?
2. A thermometer, reading $10^\circ\mathrm{C}$, is brought into a room whose temperature is $23^\circ\mathrm{C}$. Two minutes later the thermometer reading is $18^\circ\mathrm{C}$.
   
   1. Use Newton's Law of Cooling to model the change of the thermometer reading with time.
   2. How long will it take until the reading is practically $23^\circ\mathrm{C}$, say, $22.8^\circ\mathrm{C}$?
3. Solve the following initial value problem (IVP) and make a reasonably accurate sketch of the solution:
   \[ 2y^\prime + y^3 = 0;\quad y(0)=1 \]
   
4. Solve the following ODEs:
   
   1. $\left[\sin(y)\cos(y)+x\cos^2(y)\right]\,dx + x\,dy=0$
   2. $e^y\,\left[\sinh(x)\,dx + \cosh(x)\,dy\right]=0$
   3. $(x^2+y^2)\,dx - 2xy\,dy =0$
   4. $\left(\cos(xy)+\dfrac{x}{y}\right)\,dx + \left(1 + \dfrac{x}{y}\cos(xy)\right)\,dy=0$
5. 1. Under what conditions for the constants $A$,$B$,$C$,$D$, is the following ODE exact?
      \[(Ax+By)\,dx + (Cx+Dy)\,dy=0\]
   2. Solve this exact ODE.
6. 1. Solve the ordinary differential equation: $y^\prime\tan(x)=2y-8$.
   2. Sketch the family of solutions given by the general solution to the above ODE.
   3. Give the particular solution to the above ODE such that $y=0$ when $x=\pi/2$.

Precalculus - Assignment 1

Submit by: September 12

(Wait a few seconds for the images of the formulas to load)

1. Let $f:A\to B$ and $g:B\to A$ such that $g\circ f =\mathrm{id}_A$. State whether the following conclusions are valid. For valid conclusions give a proof. For invalid conclusions give a counterexample.
   
   1. $f$ is a bijection.
   2. $f$ is onto.
   3. $f$ is one-one.
   4. $g$ is a bijection.
   5. $g$ is onto.
   6. $g$ is one-one.
2. Let $A$ and $B$ be sets with 3 and 4 elements respectively. Can there be a bijection between these sets?
   
3. Exhibit a bijection from the set of even natural numbers $2\mathbb N$ to the set of all natural numbers $\mathbb N$.
   
4. Exhibit a bijection from $\mathbb N$ to the set of integers $\mathbb Z$.
   
5. Are the following functions one-one or onto?
   
   1. $f:[0,1]\to[a,b]$, $f(x)=bx+(1-x)a$.
   2. $f:\mathbb R\to\mathbb R$, $f(x)=x^2+x+1$.
   3. $f:\mathbb R\to\mathbb R$, $f(x)=x+|x|$.
6. Let $f:\mathbb N\to A$ and $g:\mathbb N\to B$ be surjective. Show there is a surjective map $h:\mathbb N\to A\cup B$.
   
7. Let $f:\mathbb R^2\to\mathbb R$ be defined by $f(x,y)=xy$. What are $f^{-1}(r)$ for $r\in\mathbb R$ and $f^{-1}([a,b])$? Draw pictures of these inverse images.
   
8. Let $f:X\to Y$. Show that
   
   1. $f$ is onto iff $f(f^{-1}(B))=B$ for every $B\subset Y$.
   2. $f$ is one-one iff $f^{-1}(f(A))=A$ for every $A\subset X$.

The Day Before You Came

The SNU hostel on August 16 - one day before the students & parents reached campus for the start of the first semester. Luckily it did not rain further and on the 17th and 18th the campus showed a much more respectable appearance. Thanks to an army of workers and pumps!

ODE Syllabus

The syllabus for the Ordinary Differential Equations being taught to the first-year B.Tech. students:

Ordinary Differential Equations of First Order: Nature of ordinary differential equations, Modeling engineering systems as differential equations, Solution methods, Applications to  law of natural  growth and decay problems, Newton’s  law of cooling,  Chemical Reactions and Solutions, Orthogonal Trajectories, Linear Equations and Non- Linear Equations, R-L circuits with step unit,  R-L Circuits with input.

Ordinary Differential Equations of Second Order: Formulation, Modeling engineering systems as second order ODEs, Conversion of some models to Differential Equations of second and higher order, Homogeneous equations (Complementary Function), Non-Homogeneous equations (Particular Integral).

Applications of Second and Higher Order ODEs: Cauchy’s linear equation, Legendre’s linear equation, Method of variation of parameters, Solving system of simultaneous ODEs. L-C-R Circuits with and without e.m.f., Oscillations of a system with damping or forcing, Oscillation and deflection of beams.

Laplace Transforms: Laplace transforms of some standard functions, properties of Laplace transforms - Linearity, First Shifting Property, Change of Scale Property, Transforms of derivatives & integrals, multiplication by $t^n$, division by $t$, Inverse  Laplace transforms,  Convolution theorem, transforms of periodic functions and Unit-Step function. Applications: Solving ODE using Laplace transforms method.

Matrices: Linear independence and dependence of a set of vectors,  Eigenvalues and eigenvectors, Stability of a system of ODEs by eigenvalues,  Orthogonality of  eigenvectors, Complex matrices, Quadratic forms and canonical forms, Diagonalization

Text Books:

1.    Advanced Engineering Mathematics by Erwin Kreyszig, Wiley India, 8^th Edition, 2006.

2.    Advanced Engineering Mathematics by Michael D. Greenberg, Pearson Education, 4^th Edition, 2008.

References:

1.    Advanced Engineering Mathematics by Alan Jeffrey, Elsevier, 2010.

2.    Higher Engineering Mathematics by B.V. Ramana, McGraw Hill Co., 2010.

3.    Engineering Mathematics, by Anthony Croft, Robert Davison, Martin Hargreaves, Pearson Education, 3^rd Edition, 2009

The First Quiz

Students of the Ordinary Differential Equations course had their first quiz on Monday, August 29, 2011 - exactly one week into their first semester. There was some consternation till it was pointed out that an early quiz is necessarily an easy quiz.

Here are the questions and their solutions:

1. Give the order and degree of $ 2x^2y^{\prime\prime} -3y^\prime +y=0$.
   
   Solution: The highest order derivative present is of second order ($y^{\prime\prime}$), so the ODE has order 2. The highest order derivative is present with degree 1, so the ODE has degree 1.
   
   
2. 
   
   
   1. Verify that $y(x)= ce^{-x}+2$ is a general solution to $y^\prime +y =2$.
      
      Solution: We calculate $y^\prime = -ce^{-x}$. Now we substitute $y$ and $y^\prime$ into the ODE:
      \[ LHS = y^\prime +y = -ce^{-x} + ce^{-x}+2 = 2 = RHS\]
      So the ODE is satisfied and the given $y$ is a general solution.
   2. Find the particular solution to this ODE given that $y=3.2$ when $x=0$.
      
      Solution: Substitute this pair of values into the general solution:
      \[ce^{-0} + 2 = 3.2\]
      We solve this for $c$ and get $c=1.2$. So the particular solution for the given initial condition is $y= 1.2 e^{-x}+2$.
      
   3. Graph this solution.
      
      Solution: