2017年11月6日月曜日

学習環境

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。


    1. y ' ' - y = 0 X 2 - 1 = 0 X = ± 1

      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 、

      y = C 1 e - 1 x + C 2 e 1 x = C 1 e - x + C 2 e x

    2. y ' ' + y = 0 X 2 + 1 = 0 X 2 = - 1 X = ± i

      よって2次方程式は虚数解0± i をもっので一般解は

      y = e 0 x ( C 1 sin 1 x + C 2 cos 1 x = C 1 sin x + C 2 cos x

    3. X 2 - 6 X + 9 = 0 X - 3 2 = 0 X = 3

      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 3x + C 2 e 3x

    4. y ' ' + π y = 0 X 2 + π = 0 X 2 = - π X = ± i π

      よって2次方程式は虚数解をもつので一般解は

      y = e 0 C 1 sin π x + C 2 cos π x = C 1 sin π x + C 2 cos π x

      初期条件より

      π = C 1 sin 0 + C 2 cos 0 C 2 = π y ' = π C 1 cos π x - π C 2 sin π x π 2 = π C 1 cos 0 - π C 2 sin 0 C 1 = π y = π sin π x + cos π x

    5. X 2 + X - 2 = 0 x + 2 X - 1 = 0 X = - 2 , 1 y = C 1 e - 2 x + C 2 e x 3 = C 1 + C 2 y ' = - 2 C 1 e - 2 x + C 2 e x - 3 = - 2 C 1 + C 2 6 = 3 C 1 C 1 = 2 C 2 = 1 y = 2 e - 2 x + e x

    6. y ' ' + 4 y ' + 13 y = 0 X 2 + 4 X + 13 = 0 X = - 2 ± 4 - 13 = - 2 ± 3 i y = e - 2 x C 1 sin 3 x + C 2 cos 3 x - 1 = C 1 sin 0 + C 2 cos 0 C 2 = - 1 y ' = - 2 e - 2 x C 1 sin 3 x + C 2 cos 3 x + e - 2 x 3 C 1 cos 3 x - 3 C 2 sin 3 x 8 = - 2 C 2 + 3 C 1 C 1 = 2 y = e - 2 x 2 sin 3 x - cos 3 x

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();








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