diff options
| author | Thomas Albers Raviola <thomas@thomaslabs.org> | 2024-05-21 11:32:09 +0200 |
|---|---|---|
| committer | Thomas Albers Raviola <thomas@thomaslabs.org> | 2024-05-21 11:32:09 +0200 |
| commit | 6fb1cd644308ac46a71536cc7da3a16377107b4f (patch) | |
| tree | 4aa8f904ced791c3b3076d78525636ef0312e6d3 /test | |
| parent | f19998f7fd9db2bd1ed4eb80ea1744a013b166fa (diff) | |
Diffstat (limited to 'test')
| -rw-r--r-- | test/test1.chn | 14 | ||||
| -rw-r--r-- | test/test1.html | 8 | ||||
| -rw-r--r-- | test/test2.chn | 19 | ||||
| -rw-r--r-- | test/test2.html | 16 | ||||
| -rw-r--r-- | test/test3.chn | 9 | ||||
| -rw-r--r-- | test/test4.chn | 65 |
6 files changed, 131 insertions, 0 deletions
diff --git a/test/test1.chn b/test/test1.chn new file mode 100644 index 0000000..f55fb70 --- /dev/null +++ b/test/test1.chn @@ -0,0 +1,14 @@ +{@define}{@author}{Thomas Albers Raviola}% +{@define}{@today}{May 12, 2024}% +{@define}{@printdate}{{@lambda}{@date}{the date is @date}}% +{@section}{Introduction} + +Hello, this is a sample of a program written in chains + +Chains was created by @author to help him write articles of questionable +usefulness. + +{@printdate}{@today} +{@section}[Conclusion] + +idk man, guess it works, don't it? diff --git a/test/test1.html b/test/test1.html new file mode 100644 index 0000000..5da418f --- /dev/null +++ b/test/test1.html @@ -0,0 +1,8 @@ +<section><h1>Introduction</h1> +Hello, this is a sample of a program written in chains +Chains was created by Thomas Albers Raviola to help him write articles of questionable +usefulness. +the date is May 12, 2024 +</section><section><h1>Conclusion</h1> +idk man, guess it works, don't it? +</section>
\ No newline at end of file diff --git a/test/test2.chn b/test/test2.chn new file mode 100644 index 0000000..a637b61 --- /dev/null +++ b/test/test2.chn @@ -0,0 +1,19 @@ +{@define}{@html-element}{{@lambda}{@name}{{@lambda}{@attr @content}{<@name @attr> +@content +</@name>}}}% +% +% +{@define}{@html-html}{{@html-element}{html}}% +{@define}{@html-head}{{@html-element}{head}}% +{@define}{@html-body}{{@html-element}{body}}% +{@define}{@html-title}{{@html-element}{title}}% +{@define}{@html-h1}{{@html-element}{h1}}% +{@define}{@html-p}{{@html-element}{p}}% +% +% +{@html-html}{}{% +{@html-head}{}{{@html-title}{}{Website}}% + +{@html-body}{}{ +{@html-h1}{}{Chains test} +{@html-p}{}{Hello World!}}} diff --git a/test/test2.html b/test/test2.html new file mode 100644 index 0000000..98a250a --- /dev/null +++ b/test/test2.html @@ -0,0 +1,16 @@ +<html > +<head > +<title > +Website +</title> +</head> +<body > + +<h1 > +Chains test +</h1> +<p > +Hello World! +</p> +</body> +</html> diff --git a/test/test3.chn b/test/test3.chn new file mode 100644 index 0000000..2fbdeab --- /dev/null +++ b/test/test3.chn @@ -0,0 +1,9 @@ +@define{@define-function}{@syntax{@name @args @body}{@define{@name}{@lambda{@args}{@body}}}}% +@define-function{@foo}{@a @b}{hola @b}% +@foo{mundo}{casa} +@define{@section}{@lambda*{@title}{@gp{@quote{@section}}}{@title@siblings@title +}}% +@section{Hello World} +hola +@section{Good Bye World} +chao diff --git a/test/test4.chn b/test/test4.chn new file mode 100644 index 0000000..e126091 --- /dev/null +++ b/test/test4.chn @@ -0,0 +1,65 @@ +{@define}{@title}{Method for solving first order and Bernoulli's differential equations} +{@define}{@author}{Thomas Albers Raviola} +{@define}{@date}{2022-10-01} +% +{@template} +{@section}{History} +I came across the method concerning this article in an old math book from Doctor +Granville (Elements of differential and integral calculus - ISBN-13: +978-968-18-1178-5). It doesn't appear to be a popular technique as when using it +for my assignments I always had to explain what I was doing. As of yet, I still +haven't found another text referencing it, which is why I decided to include it +in my website. + +In the original book this procedure is shown but never really explained, it is +left as a sort of "it just works" thing. Here is my attempt to it clear. + +{@section}{Theory} +Throughout this article we'll consider first order differential equations with +function coefficients just as a special case of the Bernoulli's differential +equation with {@eq*}{n = 0}. + +Consider now the following ODE: +{@equation}{ +y' + P(x)y = Q(x)y^n +} + +let {@eq*}{y} be the product of two arbitrary functions {@eq*}{w} and {@eq*}{z} +such that + +{@equation}{ +y &= wz \\ +y' &= w'z + wz' +} + +we now restrict {@eq*}{z} to be the solution of the ODE + +{@equation}{ +z' + P(x)z = 0 +} + +with this it is possible to solve for {@eq*}{z} by integrating + +{@equation}{ +\frac{z'}{z} = - P(x) +} + +using {@eq}{z} we solve for {@eq}{w} by replacing {@eq}{y} inside the original +ODE + +{@align}{ +w'z + wz' + P(x)wz &= Q(x)w^nz^n \\ +w'z + w\left(z' + P(x)z\right) &= Q(x)w^nz^n \\ +w'z &= Q(x)w^nz^n \\ +\frac{w'}{w^n} &= Q(x)z^{n-1} +} + +the general solution to our original ODE can be simply obtained by multiplying +{@eq*}{w} and {@eq*}{z}. + +{@section}{Comments} +This method, while functional, may not always be the most practical. In some +cases the differential equations for $w$ and $z$ may not have closed algebraic +solutions. A more traditional substitution may in some situations also be easier +than this method. Like always it is up to one to know which tool to apply for a +given problem. |