BC Calc: Journal 10

I recommend downloading the PDF; the Google Docs viewer really doesn’t do it any justice.

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\begin{flushright}
Calvin Li \\
Hawes B° \\
\today \\
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\begin{center}
\large{Journal 10: Differential equations}
\end{center}

\textit{Consider the differential equation}
\[ \frac{dy}{dx} = -\frac{2x}{3y} \]
\textit{given $(2, 2)$.}

\section{Geometric}
The slope field of this differential equation is shown, with the solution passing through $(2, 2)$ plotted. \footnote{technically, using a Runge--Kutta 4th order algorithm, not Euler's method, but the difference is small}

\begin{center}
\includegraphics[width=4in]{diff_eq.pdf}
\end{center}

Once the slope field is drawn, a geometric solution (essentially, Euler's method done geometrically rather than numerically) can be found by starting at $(2, 2)$ and following the arrows.

From this we can see that at $x=1$, $y \approx 2.45$.

\section{Euler's method}
Euler's method is a numeric form of the geometric solution: given a step $dx$ and a point $(x, y)$, a value for $dy$ can be found directly from the differential equation. $y+dy$ is then the new point; this is then iterated.

Table \ref{tab:table1} shows some values calculated by this method.
\begin{table}[t]
\centering
    \begin{tabular}{ l | l | l }
        x   & y     & dy     \\ \hline
0.600 & 2.582 & 0.031 \\
0.800 & 2.540 & 0.042 \\
1.000 & 2.487 & 0.054 \\
1.200 & 2.421 & 0.066 \\
1.400 & 2.341 & 0.080 \\
1.600 & 2.246 & 0.095 \\
1.800 & 2.133 & 0.113 \\
2.000 & 2.000 & 0.133 \\
0.400 & 2.613 & -0.020 \\
0.600 & 2.593 & -0.031 \\
0.800 & 2.562 & -0.042 \\
1.000 & 2.520 & -0.053 \\
1.200 & 2.467 & -0.065 \\
1.400 & 2.403 & -0.078 \\
1.600 & 2.325 & -0.092 \\
1.800 & 2.233 & -0.107 \\
2.000 & 2.126 & -0.125 \\
2.200 & 2.000 & -0.147 \\
2.400 & 1.854 & -0.173 \\
2.600 & 1.681 & -0.206 \\
2.800 & 1.475 & -0.253 \\
3.000 & 1.222 & -0.327 \\
    \end{tabular}
\caption{Approximations of the solution using Euler's method.}
\label{tab:table1}
\end{table}
From it we can see that at $x = 1.000$, $y \approx 2.487$ using our method.
\section{Algebraic solution}
The differential equation can also be solved algebraically:
\begin{align*}
\frac{dy}{dx} &= \frac{-2x}{3y} \\
3y ~dy &= -2x ~dx \\
\int 3y ~dy &= \int -2x ~dx \\
\frac{3}{2} y^2 &= -x^2 + c \\
\frac{3 y^2}{2} + x^2 &= c
\end{align*}
From this \textit{general solution} we can see that the solution is an ellipse. Knowing that $y = 2$ when $x = 2$, we can find a \textit{particular solution}:
\begin{align*}
\frac{3 \times 2^2}{2} + 2^2 &= c \\
\frac{3 y^2}{2} + x^2 &= 10
\end{align*}
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\end{document}

Spanish strong/weak vowels

The so-called “strong vowels” (vocales fuertes) in Spanish are ae, and o. The “weak vowels” (vocales débiles) are u and i.

Compare the IPA vowel classification:

IPA vowel chart. See http://en.wikipedia.org/wiki/File:IPA_vowel_chart_2005.png.

“Strong” and “weak” evidently correspond to IPA “open” and “close”.

The rise of single-board computers

I’m not sure if you’ve noticed, but there seems to be a boom in so-called “single-board computers”—to wit, the ArduinoRaspberry PiParallella, and Electric Imp. Products like these have, I think, the capability to finally achieve ubiquitous computing.

Arduino

I think most people (or at least, most of the people interested in this field) know about the Arduino, so I won’t discuss it too much here, except to note that it’s pretty hardware-focused.

Raspberry Pi

The Raspberry Pi is actually a fully-functional computer—it runs a Debian variant and has video (HDMI) output and USB input. It’s in the roughly same form factor as the Arduino. The Raspberry Pi Foundation sees it as a way to teach comp-sci to kids—that’s certainly not the first thing I think of. It’s definitely more software-oriented than the Arduino—in fact, I’m not sure the hardware is that hackable at all….

Parallella

Parallella got a bit of attention from its Kickstarter project. I have no idea if their claims match reality (it’s certainly novel, and sounds far-fetched, but seems to be legitimate), but it is a very neat idea. Supercomputing power in the hands of the masses (or at least, the interested laity) can only be good for the world. Here’s to hoping they can get this off the ground.

Electric Imp

The Electric Imp is fascinating—a basic CPU and WiFi chip, in the form factor of an SD card. It wasn’t that long ago that I was impressed by computers in the form factor of a wall wart. And here we have what is essentially a computer, the size of an SD card. It basically does roughly what you’d expect out of an Arduino. Or perhaps slightly less—if I have it correctly, the Arduino seems to have taken on a role as a general-purpose embedded/mobile microcontroller. The Electric Imp seems to aspire to somewhat less—although the Imp targets more the ongoing trend of The Internet of Things, something the Arduino does not, which gives the Imp a benefit. Yet the Imp has not yet caught on nearly as much as the Arduino or Raspberry Pi. Here’s to hoping it does.

One thing I’ve noticed is that there’s a lot of flak about how the Imp is controlled by their servers (cloud control). I share the concerns, but I think the cloud concept is probably in the end the best solution.

Cyrus Farivar over at Ars Technica apparently took it out for a spin.

The company behind it also happens to be local (Los Altos).

 

Exciting times.