Topics

646

Ch 4 / Bonus problem for week 5a

« on: February 07, 2013, 11:52:36 PM »

Write down a 2nd order homogeneous linear equation such that it has solutions
\begin{equation*}
y_1= e^t, \qquad y_2= te^{-t}.
\end{equation*}

647

Technical Questions / LaTeX challenge

« on: February 07, 2013, 08:01:30 AM »

Using LaTeX (NOT plain TeX) produce a document in 14 pts (like your Quizzes and soon to be written Tests).

648

General Discussion / Final exams

« on: February 07, 2013, 04:43:07 AM »

Please remember that Final Exams are conducted and regulated by Faculty of Arts and Science and instructors cannot provide any additional sittings, resolve conflicts or accept any medical notes covering it. See rules:

http://www.artsandscience.utoronto.ca/academichandbook.shtml#3.4

649

General Discussion / Previous years resources

« on: February 01, 2013, 05:08:52 AM »

I used to run them (MAT244 forums and wikis for 2010, 2011, 2012 and APM346 forum and wiki for 2011) on my office computer. I replaced my office computer by a newer one this Wednesday (Jan 30, 2013) and moved all the staff to this new computer changing configurations.

With (each) wiki it was straightforward: move files, change LocalSettings.php and export/import MySQL database.

With (each) forum it was more tedious and error prone as many settings are stored not in configuration files but in database which I went through editing. I could miss something (hope not!), so if you notice any problems there -- missed icons, avatars, smileys, attachments, bad links--advise me!

650

Ch 3 / Problem of the week 4b

« on: January 31, 2013, 05:50:11 PM »

Consider two identical and connected harmonic oscillators:
\begin{equation}
\left\{\begin{aligned}
&y''+K y + L (y-z)=0\\
&z''+Kz + L(z-y)=0
\end{aligned}\right.
\label{eq-1}
\end{equation}
with $K>0$, $L>0$.

Even if this is a system one can add or subtract equations getting system of two equations describing $y+z$ and $y-z$ separately.

1) Find $y+z$ and $y-z$ and then $y$ and $z$  (so, find the general solution of (\ef{eq-1}).

2) What frequencies has the described system?

651

Ch 3 / Problem of the week 4a

« on: January 31, 2013, 05:35:09 PM »

Consider equation describing harmonic oscillations
\begin{equation}
y'' + \omega^2 y=0
\label{eq-1}
\end{equation}
and add damping
\begin{equation}
y'' + \alpha y'+ \omega^2 y=0,\qquad \alpha>0.
\label{eq-2}
\end{equation}

a) Describe solutions

Now add external force
\begin{equation}
y'' + \alpha y'+ \omega^2 y=\cos(\beta t)
\label{eq-3}
\end{equation}
and find solution in the form
\begin{equation}
 y=A(\beta ) \cos(\beta t)+B(\beta )\sin(\beta t).
\label{eq-4}
\end{equation}

b) Find resonance frequency $\beta$ such that amplitude $C(\beta)=\sqrt{A(\beta)^2+B(\beta)^2}$ reaches maximum.

652

Quiz 2 / Day Section, Question 1

« on: January 31, 2013, 01:29:59 PM »

Determine the values of $\alpha$, if any, for which all solutions of the following ODE tend to zero as $t\to\infty$ as well as all values of $\alpha$, if any, for which all nonzero solutions become unbounded as $t\to\infty$
$$
y'' - (2\alpha-1)y'+\alpha(\alpha-1)y=0.
$$

653

Quiz 2 / Night sections, 3.2 p 156, # 17

« on: January 30, 2013, 08:06:44 PM »

If the Wronskian $W=W(f,g)$ of two functions $f$ and $g$ is equal to $3e^{4t}$ and if $f=e^{2t}$, find $g(t)$.

654

Quiz 2 / Night Sections, 2.6 p 102, # 25

« on: January 30, 2013, 08:05:38 PM »

Find an integrating factor and solve the given equation
\begin{equation*}
(3x^2y+2xy+y^3)+(x^2+y^2)y'=0
\end{equation*}

655

Technical Questions / Bad LaTeX

« on: January 25, 2013, 01:47:17 AM »

I noticed constructions like

Code: [Select]

$a\\ b$ and

Code: [Select]

$$a\\ b$$
This is not good! Reason: the former works in LaTeX but here \\ means linebreak and in inline math here you get $1\\2$ (improper vertical alignment) and in the real LaTeX 2 will just move to the beginning of the next line.

The latter does not work at all (simply ignored) as LaTeX has several multi-lined environments: gather, split, align and multline (in AMS-LaTeX which we support here), their * versions gather*, split*, align* and multline*  and also gathered, aligned.

As we want to be able to copy-paste from MathJax to LaTeX with no editing we should use a proper syntax.

PS. Opposite is not true as MJ is not full LaTeX environment, and is not intended to be. Due to technical reasons (inner mathml engine) even \intertext does not work in MJ.

PPS. There are lots of things like it works but is deprecated

656

Ch 3 / Bonus problem for week 3b

« on: January 24, 2013, 09:56:47 AM »

(a). Reduce to the first order equation by substitution $y'=z$ and then find the general solutions (solutions using characteristic roots will not be considered)

\begin{equation}
y'' + \omega^2 y=0
\end{equation}


(b)  Also solve initial value problem
$y(0)=1$, $y'(0)=0$.

657

Ch 3 / Bonus problem for week 3a

« on: January 24, 2013, 09:56:29 AM »

(a). Reduce to the first order equation by substitution $y'=z$ and then find the general solutions (solutions using characteristic roots will not be considered)

\begin{equation}
y'' -\omega^2 y=0
\end{equation}


(b)  Also solve initial value problem
$y(0)=1$, $y'(0)=0$.

658

Ch 1--2 / Handouts

« on: January 23, 2013, 02:06:01 PM »

Three handouts placed on Handouts page (duplicated on
Home Assignments page.

659

Quiz 1 / Day section 2.6 #25

« on: January 21, 2013, 11:05:13 AM »

Please post solution:

Find an integrating factor and solve the given equation
$$
(3x^2y + 2xy + y^3) + (x^2 + y^2)y '= 0.
$$

660

Quiz 1 / Day section 2.1 #18

« on: January 21, 2013, 11:01:25 AM »

Please post solution:

Find the solution of the given initial value problem:
$$
ty' + 2y = \sin (t) \quad  (t > 0),\qquad  y(\pi/2) = 1.
$$