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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

Scroll through the document using the arrows at the base of the page.

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!