数学 - Python - JavaScript - 面積、体積、長さ - 積分法の応用 – 簡単な微分方程式 - 2階微分方程式(定数係数の2階線形同次微分方程式の一般解、場合分け、初期条件、定数)

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)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo(iPad アプリ)
• 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)、2階微分方程式、問49.を取り組んでみる。

49. 1. 
       $\begin{array}{}y\text{'}\text{'}-y=0\\ X^2-1=0\\ X=\pm \; <!--\; plus-minus\; sign\; -->1\end{array}$

       よって2次方程式は異なる2つの実数解-1、1をもつので一般解は 、

       $\begin{array}{}y=C_1{e}^{-1x}+C_2{e}^{1}x\\ =C_1{e}^{-x}+C_2{e}^x\end{array}$
    2. 
       $\begin{array}{}y\text{'}\text{'}+y=0\\ X^2+1=0\\ X^2=-1\\ X=\pm \; <!--\; plus-minus\; sign\; -->i\end{array}$

       よって2次方程式は虚数解0± i をもっので一般解は

       $\begin{array}{}y=e^{0x}(C_1\sin 1x+C_2\cos 1x\\ =C_1\sin x+C_2\cos x\end{array}$
    3. 
       $\begin{array}{}{X}^2-6X+9=0\\ {\left(X-3\right)}^2=0\\ X=3\end{array}$

       よって2次方程式は重解をもつので一般解は

       $y=C_{1}x{e}^{\mathrm{3x}}+C_2{e}^{\mathrm{3x}}$
    4. 
       $\begin{array}{}y\text{'}\text{'}+\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}y=0\\ X^2+\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}=0\\ X^2=-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\\ X=\pm \; <!--\; plus-minus\; sign\; -->i\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\end{array}$

       よって2次方程式は虚数解をもつので一般解は

       $\begin{array}{}y=e^0\left(C_1\sin \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x+C_2\cos \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x\right)\\ =C_1\sin \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x+C_2\cos \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x\end{array}$

       初期条件より

       $\begin{array}{}\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}=C_1\sin 0+C_2\cos 0\\ C_2=\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\\ y\text{'}=\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}{C}_1\cos \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}{C}_2\sin \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x\\ {\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}}^2=\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}{C}_1\cos 0-\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}{C}_2\sin 0\\ C_1=\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\\ y=\mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}\left(\sin \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x+\cos \mathrm{\pi \; <!--\; greek\; small\; letter\; pi\; -->}x\right)\end{array}$
    5. 
       $\begin{array}{}{X}^2+X-2=0\\ \left(x+2\right)\left(X-1\right)=0\\ X=-2,1\\ y=C_1{e}^{-2x}+C_2{e}^x\\ 3=C_1+C_2\\ y\text{'}=-2C_1{e}^{-2x}+C_2{e}^x\\ -3=-2C_1+C_2\\ 6=3C_1\\ C_1=2\\ C_2=1\\ y=2e^{-2x}+e^x\end{array}$
    6. 
       $\begin{array}{}y\text{'}\text{'}+4y\text{'}+13y=0\\ X^2+4X+13=0\\ X=-2\pm \; <!--\; plus-minus\; sign\; -->\sqrt{4-13}\\ =-2\pm \; <!--\; plus-minus\; sign\; -->3i\\ y=e^{-2x}\left(C_1\sin 3x+C_2\cos 3x\right)\\ -1=C_1\sin 0+C_2\cos 0\\ C_2=-1\\ y\text{'}=-2e^{-2x}\left(C_1\sin 3x+C_2\cos 3x\right)+e^{-2x}\left(3C_1\cos 3x-3C_2\sin 3x\right)\\ 8=-2C_2+3C_1\\ C_1=2\\ y=e^{-2x}\left(2\sin 3x-\cos 3x\right)\end{array}$

コード(Emacs)

Python 3

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

x, C1, C2 = symbols('x, C1, C2')
y = C1 * exp(-x) + C2 * exp(x)
eq = Derivative(y, x, 2) - y

for t in [y, eq, eq.doit() == 0]:
    pprint(t)

入出力結果(Terminal, Jupyter(IPython))

$ ./sample49.py
    -x       x
C₁⋅ℯ   + C₂⋅ℯ 
                     2                
      -x       x    ∂ ⎛    -x       x⎞
- C₁⋅ℯ   - C₂⋅ℯ  + ───⎝C₁⋅ℯ   + C₂⋅ℯ ⎠
                     2                
                   ∂x                 
True
$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="0" value="1">
<label for="ε0">ε = </label>
<input id="ε0" type="number" min="0" value="0.1">
<label for="a0">A = </label>
<input id="a0" type="number" value="1">
<label for="α0">α = </label>
<input id="α0" 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="sample49.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'),
    input_n0 = document.querySelector('#n0'),
    input_ε0 = document.querySelector('#ε0'),
    input_a0 = document.querySelector('#a0'),
    input_α0 = document.querySelector('#α0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_n0, input_a0, input_α0],
    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 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),
        n0 = parseFloat(input_n0.value),
        ε0 = parseFloat(input_ε0.value),
        a0 = parseFloat(input_a0.value),
        α0 = parseFloat(input_α0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }
    
    let points = [],
        lines = [],
        σ = Math.sqrt(n0 ** 2 - ε0 ** 2),
        f1 = (x) => Math.exp(-ε0 * x) * Math.sin(σ * x + α0),
        f2 = (x) => Math.exp(-ε0 * x) * Math.cos(σ * x + α0),
        g = (x) => a0 * Math.exp(- ε0 * x),
        h = (x) => -g(x),
        fns = [[f1, 'red'],
               [f2, 'green'],
               [g, 'blue'],
               [h, '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 =
n = ε = A = α =