数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 簡単な微分方程式 - 解曲線と初期条件(累乗(べき乗)、指数関数、三角関数(正弦)、一般解、任意定数、特殊解)

数学読本〈5〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

数学読本〈5〉微分法の応用/積分法/積分法の応用/行列と行列式(松坂 和夫(著)、岩波書店)の第20章(面積、体積、長さ - 積分法の応用)、20.4(簡単な微分方程式)、解曲線と初期条件、問32.を取り組んでみる。

32. 1. • 解。

         $y=\frac{1}{3}{x}^3+C$

       • 特殊解。

         $\begin{array}{l}2=\frac{1}{3}·0^3+C\\ C=3\\ y=\frac{1}{3}{x}^3+2\end{array}$

    2. • 解。

         $y=C{e}^{3x}$

       • 特殊解。

         $\begin{array}{l}5=C{e}^{3·0}\\ C=5\\ y=5e^{3x}\end{array}$

    3. • 解。

         $y=-\frac{1}{2}\cos 2x+C$

       • 特殊解。

         $\begin{array}{l}0=-\frac{1}{2}\cos 2\pi +C\\ 0=-\frac{1}{2}·1+C\\ C=\frac{1}{2}\\ y=-\frac{1}{2}\cos 2x+\frac{1}{2}\\ =\frac{1}{2}\left(-\cos 2x+1\right)\end{array}$

    各Cは定数。

Cはそれぞれ任意定数(積分定数)。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Rational, exp, cos, sin, Derivative, solve, pi, plot

print('32.')
x, C = symbols('x C')
ys = [(Rational(1, 3) * x ** 3 + C, x ** 2, 0, 2),
      (C * exp(3 * x), 3 * C * exp(3 * x), 0, 5),
      (- Rational(1, 2) * cos(2 * x) + C, sin(2 * x), pi, 0)]

for i, (y, y1, x0, y0) in enumerate(ys, 1):
    print(f'({i})')
    D = Derivative(y, x, 1)
    for t in [D, D.doit()]:
        pprint(t)
        print()
    print(D.doit() == y1)
    s = solve(y0 - y.subs({x: x0}))
    for t in [s, y.subs({C: s[0]})]:
        pprint(t)
        print()
    p = plot(y.subs({C: s[0]}), y1.subs({C: s[0]}), show=False, legend=True)
    for j, color in enumerate(['red', 'green']):
        p[j].line_color = color
    p.save(f'sample32_{i}.svg')
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample32.py
32.
(1)
  ⎛     3⎞
∂ ⎜    x ⎟
──⎜C + ──⎟
∂x⎝    3 ⎠

 2
x 

True
[2]

 3    
x     
── + 2
3     


(2)
∂ ⎛   3⋅x⎞
──⎝C⋅ℯ   ⎠
∂x        

     3⋅x
3⋅C⋅ℯ   

True
[5]

   3⋅x
5⋅ℯ   


(3)
∂ ⎛    cos(2⋅x)⎞
──⎜C - ────────⎟
∂x⎝       2    ⎠

sin(2⋅x)

True
[1/2]

  cos(2⋅x)   1
- ──────── + ─
     2       2


$

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">

<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="sample32.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    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'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    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 f1 = (x) => 1 * Math.exp(3 * x),
    f2 = (x) => 5 * Math.exp(3 * x),
    f3 = (x) => 10 * Math.exp(3 * x);

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);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [[0, y1, 0, y2, 'red'],
                 [x1, 5, x2, 5, 'red']],
        fns = [[f1, 'green'],
               [f2, 'blue'],
               [f3, 'orange']],
        fns1 = [],
        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])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    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();

r = dx =
x1 = x2 =
y1 = y2 =