## 2017年10月24日火曜日

### 数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 簡単な微分方程式 - 曲線群に共通な性質を表す微分方程式(累乗(べき乗)、逆数、円)

1. $y\text{'}=2x$

2. $\begin{array}{l}y\text{'}=2Cx\\ y\text{'}x=2C{x}^{2}\\ y\text{'}x=2y\end{array}$

3. $\begin{array}{l}y\text{'}=\frac{-1}{{\left(x+C\right)}^{2}}\\ =-{y}^{2}\end{array}$

4. $\begin{array}{l}2x+2yy\text{'}=0\\ yy\text{'}=-x\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Derivative, solve, sqrt

print('38.')
x, y, C = symbols('x y C')

ys = [x ** 2 + C,
C * x ** 2,
1 / (x + C),
sqrt(C - x ** 2)]

for i, y0 in enumerate(ys, 1):
print(f'({i})')
D = Derivative(y0, x, 1)
C0 = solve(y - y0, C)[0]
for t in [D, D.doit(), D.doit().subs({C: C0})]:
pprint(t)
print()
print()

$./sample38.py 38. (1) ∂ ⎛ 2⎞ ──⎝C + x ⎠ ∂x 2⋅x 2⋅x (2) ∂ ⎛ 2⎞ ──⎝C⋅x ⎠ ∂x 2⋅C⋅x 2⋅y ─── x (3) ∂ ⎛ 1 ⎞ ──⎜─────⎟ ∂x⎝C + x⎠ -1 ──────── 2 (C + x) 2 -y (4) ⎛ ________⎞ ∂ ⎜ ╱ 2 ⎟ ──⎝╲╱ C - x ⎠ ∂x -x ─────────── ________ ╱ 2 ╲╱ C - x -x ─────── ____ ╱ 2 ╲╱ y$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="c0">C = </label>
<input id="c0" type="number" value="1">
<label for="x0">x0 = </label>
<input id="x0" type="number" value="2">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample38.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
pre0 = document.querySelector('#output0'),
width = 600,
height = 600,
btn0 = document.querySelector('#draw0'),
btn1 = document.querySelector('#clear0'),
input_r = document.querySelector('#r0'),
input_dx = document.querySelector('#dx'),
input_x1 = document.querySelector('#x1'),
input_x2 = document.querySelector('#x2'),
input_y1 = document.querySelector('#y1'),
input_y2 = document.querySelector('#y2'),
input_c0 = document.querySelector('#c0'),
input_x0 = document.querySelector('#x0'),
inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
input_c0, input_x0],
p = (x) => pre0.textContent += x + '\n',
range = (start, end, step=1) => {
let res = [];
for (let i = start; i < end; i += step) {
res.push(i);
}
return res;
};

let C = 20 * Math.exp(2),
f = (x) =>  C * Math.exp(-x / 2),
f1 = (x) => -1 / 2 * f(x),
g = (x0) => (x) => f1(x0) * (x - x0) + f(x0);

let draw = () => {
pre0.textContent = '';

let r = parseFloat(input_r.value),
dx = parseFloat(input_dx.value),
x1 = parseFloat(input_x1.value),
x2 = parseFloat(input_x2.value),
y1 = parseFloat(input_y1.value),
y2 = parseFloat(input_y2.value),
c0 = parseFloat(input_c0.value),
x0 = parseFloat(input_x0.value);

if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
return;
}

let points = [],
lines = [],
f = (x) => c0 * x ** 2,
f1 = (x) => 2 * c0 * x,
g = (x) => f1(x0) * (x - x0) + f(x0),
h = (x) => f(x0) / x0 * x,
fns = [[f, 'red']],
fns1 = [[g, 'green'],
[h, 'blue']],
fns2 = [];

fns.forEach((o) => {
let [fn, color] = o;
for (let x = x1; x <= x2; x += dx) {
let y = fn(x);

if (Math.abs(y) < Infinity) {
points.push([x, y, color]);
}
}
});
fns1.forEach((o) => {
let [fn, color] = o;

lines.push([x1, fn(x1), x2, fn(x2), color]);
});
fns2.forEach((o) => {
let [fn, color] = o;

for (let x = x1; x <= x2; x += dx0) {
let g = fn(x);

lines.push([x1, g(x1), x2, g(x2), color]);
}
});
let xscale = d3.scaleLinear()
.domain([x1, x2])
let yscale = d3.scaleLinear()
.domain([y1, y2])

let xaxis = d3.axisBottom().scale(xscale);
let yaxis = d3.axisLeft().scale(yscale);
div0.innerHTML = '';
let svg = d3.select('#graph0')
.append('svg')
.attr('width', width)
.attr('height', height);

svg.selectAll('line')
.data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
.enter()
.append('line')
.attr('x1', (d) => xscale(d[0]))
.attr('y1', (d) => yscale(d[1]))
.attr('x2', (d) => xscale(d[2]))
.attr('y2', (d) => yscale(d[3]))
.attr('stroke', (d) => d[4] || 'black');

svg.selectAll('circle')
.data(points)
.enter()
.append('circle')
.attr('cx', (d) => xscale(d[0]))
.attr('cy', (d) => yscale(d[1]))
.attr('r', r)
.attr('fill', (d) => d[2] || 'green');

svg.append('g')
.attr('transform', translate(0, ${height - padding})) .call(xaxis); svg.append('g') .attr('transform', translate(${padding}, 0))
.call(yaxis);

[fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();