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661

Ch 1--2 / Bonus problem for week 2

« on: January 17, 2013, 02:18:45 PM »

Equation
$$
y'=\frac{y-x-2}{y+x}
$$
by a change of variables $x=t+a$, $y=z+b$reduce to homogeneous equation and solve it. Express $y$ as an implicit function of $x$:
$$
F(x,y)=C.
$$

662

Quiz 1 / Night section, 2.2 # 26

« on: January 16, 2013, 07:34:46 PM »

Please post solution

Solve the initial value problem
$$y′=2(1+x)(1+y^2),\qquad y(0)=0$$
and determine where the solution attains its minimal value. -- Added later.

663

Quiz 1 / Night section: 2.1 #17

« on: January 16, 2013, 07:33:36 PM »

Please post solution:

Find solution of the given initial value problem
$$y′−2y=e^{2t},\qquad y(0)=2$$

664

Ch 1--2 / Bonus problem of the week 1:

« on: January 10, 2013, 07:34:59 AM »

Submitting (first) the correct solution of the "Problem of the Week" (I will try to post them each week) you get karma which translates to bonus marks.

• (a) Find the general solution of
  \begin{equation}
  x y'= 2 y(4-y)
  \label{eq-1}
  \end{equation}
  
• (b) Find solution of (\ref{eq-1}) (as $x>0$) satisfying initial condition $y(1)=1$;
• (c) Using computer f.e. http://math.rice.edu/~dfield/dfpp.html solve (a), (b) graphically: output will two pictures: each containing a field of directions and several integral lines in (a), and one particular line in (b).

665

Final Exam / Posting Final Exam Solutions

« on: December 20, 2012, 06:13:04 AM »

At 13:30 I will post problem conditions and only then you may post solutions as a replies to my posts. No credits for posts after Dec 25, 23:59.

666

Technical Questions / LaTeX advices

« on: November 20, 2012, 06:14:04 AM »

We are talking about LaTeX in full, not MathJax

1) Strongly recommended: \documentclass[]{memoir}--very flexible well maintained, can emulate book and article. It allows normalfontsize 14 pt \documentclass[14pt]{memoir} -- note how tests are typed for convenience

2) Strongly recommended: \usepackage{hyperref}--makes better pdf, clickable links (in \label--\ref, \bibitem-\cite, in table of content, index) and also bookmarks

3) Strongly recommended \usepackage{microtype}--better typesetting

4) For graphics (LaTeX drawings) \usepackage{pgf}--the most powerful (another pstricks is also great)

5) For presentations \documentclass{beamer}-- nearest competition powerdot

667

Term Test 2 / TT2--Problem 5

« on: November 15, 2012, 08:26:34 PM »

  Let $Q = \{(x,y)\in {\mathbb{R}}^2: |x|<1, |y|<1\}.$ Draw the set $Q.$ We define data $g$ on the boundary of $Q$:

Find the solution $u$ of the Dirichlet problem on $Q$:
\begin{equation*}
\Delta u=0 \qquad \text{for  }  (x,y) \in Q
\end{equation*}
with the boundary conditions
\begin{equation*}
u = \left\{\begin{aligned}
&y &&\text{as  }x=1,\\[3pt]
-&y &&\text{as }x=-1,\\[3pt]
&x &&\text{as   }y=1,\\[3pt]
-&x &&\text{as   }y=-1.
\end{aligned}\right.
\end{equation*}

Post after 22:30

668

Term Test 2 / TT2--Problem 4

« on: November 15, 2012, 08:23:51 PM »

Find Fourier transform of the  function
\begin{equation*}
f(x)= \left\{\begin{aligned}
&1-|x| &&|x|<1\\
&0 &&|x|>1.
\end{aligned}\right.
\end{equation*}
and write this function $f(x)$ as a Fourier integral.

Post after 22:30

669

Term Test 2 / TT2--Problem 3

« on: November 15, 2012, 08:22:48 PM »

Let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ satisfy the following conditions:

• $\phi$ is continuous.
• $\phi'$ is continuous.
• $\phi(x) = 0$ for all $|x|>1$.

Consider the integral
$$
I_\lambda = \int_{-\infty}^{+\infty} \phi (x) \cos (\lambda x)\, dx.
$$
Prove that $|I_\lambda| \rightarrow 0$ when $\lambda \rightarrow \infty$.


Post after 22:30

670

Term Test 2 / TT2--Problem 2

« on: November 15, 2012, 08:20:35 PM »

  Consider the diffusion equation
\begin{equation*}
u_t -ku_{xx}=0 \qquad \text{for} \quad t>0,\ x \in (0,2\pi)
\end{equation*}
with the boundary conditions
\begin{equation*}
u_x(0,t)=u_x(2\pi,t)=0
\end{equation*}
and the initial condition
\begin{equation*}
u(x,0)=|\sin (x)|.
\end{equation*}

• (a) Write the associated eigenvalue problem.
• (b) Find all  eigenvalues and corresponding eigenfunctions.
• (c) Show that the eigenfunctions associated to 2 different eigenvalues are orthogonal.
• (d) Write the solution in the form of  a series expansion.
• (e) Write a formula for  the coefficients of the series expansion.

Post after 22:30

671

Term Test 2 / TT2--Problem 1

« on: November 15, 2012, 08:14:38 PM »

Let $f:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be a continuous but non-differentiable function which satisfies $f(x)=0$ for all $|x| > 1.$ Let $g:{\mathbb{R}}\rightarrow {\mathbb{R}}$ be a continuous function which satisfies satisfies $g(x)=0$ for all $|x| > 2.$ Suppose further that derivative $g'$ and second derivative $g''$ are both continuous. The convolution $f*g$ of these two functions is defined by the formula
$$
(f*g) (x) = \int f(x-y)g(y)\,dy.
$$

• (a) Prove that the function $f*g(x) =0$ for $|x|>3$.
• (b) Prove that the derivative of the function $f*g$ is continuous.

post after 22:30

672

Term Test 2 / TT2 -- posting solutions

« on: November 14, 2012, 10:58:58 AM »

I will post problems Thu, Oct 15 at 20:30--20:45 and you may post solutions at 22:30.

673

Home Assignment Y / HAY--as preparation for TT2

« on: November 09, 2012, 09:07:51 AM »

http://www.math.toronto.edu/courses/apm346h1/20129/HAY.html
http://www.math.toronto.edu/courses/apm346h1/20129/HAY.pdf

You may post solutions immediately

674

Home Assignment 5 / About HA5

« on: November 01, 2012, 05:10:42 AM »

Some of subproblems are just tricks:

Example: Decompose into full F.s. $\cos (mx)$. Answer: $\cos (mx)$ as it is one of the basic functions (however it would not be so if we decompose into $\sin$-F.s., or F.s with respect to $\sin((n+\frac{1}{2}x)$, $n=0,1,\ldots$.


What is the sum F.s. on $(-\infty,\infty) was discussed in http://www.math.toronto.edu/courses/apm346h1/20129/L16.html

Again see below:

675

General Discussion / Merry Halloween

« on: October 30, 2012, 01:30:28 PM »

Two last slides from my talk in MSRI Oct 31, 2008 (it was the last talk on the conference)