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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<!-- Generated html may include non-free tasks and be non-redistributable. -->
<!-- phapl_tpl.html, template: -->
<!-- Copyright © 2018 Aleksey Cherepanov <lyosha@openwall.com> -->
<!-- Redistribution and use in source and binary forms, with or without modification, are permitted. -->
<html>
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>PhaPl: Phase Plane Helper</title>
<style>
body { font-size: 140%; }
input { width: 600px; }
.container { width: 90%; border: 1px solid black; padding: 10px; }
</style>
<script type="text/javascript" src="lib/mathjax/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
displayAlign: "left"
});
</script>
<script src="lib/lzma-d-min.js" type="text/javascript"></script>
<!-- shim for ES6 `Promise` builtin -->
<script src="lib/Promise.min.js" type="text/javascript"></script>
<!-- shim for off-main-thread function compilation -->
<script src="lib/FunctionPromise.js" type="text/javascript"></script>
<script src="lib/pypyjs.js" type="text/javascript"></script>
<!-- %% catch errors in javascript -->
<script type="text/javascript">
langtr_lang_index = 1;
phapl_cached_solutions = {};
phapl_start = 0;
function langtr_replace(s) {
return s.replace(/\[\[([^\x5d]*)\]\]/g, function (match, p1) {
return p1.split(' | ')[langtr_lang_index];
})
}
function scroll_to(id) {
document.getElementById(id).scrollIntoView({ "behaviour" : "smooth" });
}
function phapl_populate_cache(b64) {
var l = atob(b64);
var t = [];
for (var i = 0; i < l.length; i++) {
t.push(l.charCodeAt(i));
}
l = t;
var lzma = LZMA;
lzma.decompress(l,
function (result, error) {
// %% handle errors
if (error) {
console.log('error decompressing lzma phapl\'s prepopulated cache: ' + error);
}
var ll = JSON.parse(result);
for (var i in ll) {
phapl_cached_solutions[ll[i][0] ] = [ ll[i][1], ll[i][2] ];
}
},
function (percent) {});
}
function phapl_show_results(h) {
document.getElementById('thediv').innerHTML = h[0];
console.log('html ok');
eval(h[1]);
console.log('canvas ok');
MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById('thediv')]);
console.log(new Date().getTime() - phapl_start);
}
function phapl_do() {
document.getElementById('thediv').innerHTML = 'PyPy.js is starting. It may take a minute...';
phapl_start = new Date().getTime();
var dot_x = document.getElementById('dot_x').value;
var dot_y = document.getElementById('dot_y').value;
var t = '"' + dot_x + '", "' + dot_y + '"';
if (t in phapl_cached_solutions) {
phapl_show_results(phapl_cached_solutions[t]);
} else {
pypyjs.stdout = pypyjs.stderr = console.log;
pypyjs.ready().then(function() {
document.getElementById('thediv').innerHTML = 'PyPy.js started successfully. Calculations are happening...';
return pypyjs.set('dot_x', dot_x);
}).then(function() {
return pypyjs.set('dot_y', dot_y);
}).then(function() {
return pypyjs.exec(
'import traceback\n' +
'try:\n' +
' import libphapl\n' +
' r = libphapl.task_to_html(str(dot_x), str(dot_y))\n' +
'except:\n' +
' r = [ u"\\x3cpre\\x3e" + traceback.format_exc() + "\\x3c/pre\\x3e", "" ]\n'
);
}).then(function () {
console.log('python part ok');
return pypyjs.get('r');
}).then(function (h) {
h[0] = langtr_replace(h[0]);
h[1] = langtr_replace(h[1]);
phapl_cached_solutions[t] = h;
phapl_show_results(h);
});
}
}
function phapl_gen_linear(equilibria_type, start_p) {
scroll_to('phapl_div_run');
document.getElementById('thediv').innerHTML = 'PyPy.js is starting. It may take a minute...';
phapl_start = new Date().getTime();
pypyjs.stdout = pypyjs.stderr = console.log;
pypyjs.ready().then(function() {
document.getElementById('thediv').innerHTML = 'PyPy.js started successfully. Calculations are happening...';
return pypyjs.set('equilibria_type', equilibria_type);
}).then(function() {
return pypyjs.exec(
'import traceback\n' +
'try:\n' +
' import tg1\n' +
' r = tg1.gen_linear(str(equilibria_type))\n' +
'except:\n' +
' r = [ u"\\x3cpre\\x3e" + traceback.format_exc() + "\\x3c/pre\\x3e" ]\n'
);
}).then(function () {
console.log('gen: python part ok');
return pypyjs.get('r');
}).then(function (h) {
if (h.length == 1) {
document.getElementById('thediv').innerHTML = h[0];
} else {
document.getElementById('dot_x').value = h[0];
document.getElementById('dot_y').value = h[1];
phapl_reset_color();
if (start_p) {
phapl_do();
} else {
document.getElementById('thediv').innerHTML = langtr_replace(h[2]);
MathJax.Hub.Queue(["Typeset", MathJax.Hub, document.getElementById('thediv')]);
}
console.log(new Date().getTime() - phapl_start);
}
scroll_to('phapl_div_run');
});
}
function collapse(id) {
var e = document.getElementById(id);
var t = e.style.display;
if (t != 'none') {
t = 'none';
} else {
t = 'block';
}
e.style.display = t;
e = document.getElementById(id + '-header');
t = e.value.split(': ');
if (t[0].match('Show')) {
t = '\u25bc Hide tasks: ' + t[1];
} else {
t = '\u25ba Show tasks: ' + t[1];
}
e.value = t;
}
phapl_colored_td = null;
function phapl_set(obj, dotx, doty) {
if (phapl_colored_td) {
phapl_colored_td.style.backgroundColor = null;
}
obj.style.backgroundColor = 'lightgreen';
document.getElementById('dot_x').value = dotx;
document.getElementById('dot_y').value = doty;
phapl_colored_td = obj;
scroll_to('phapl_div_run');
phapl_do();
}
function phapl_reset_color() {
if (phapl_colored_td) {
phapl_colored_td.style.backgroundColor = null;
}
}
function phapl_setup_canvases(phapl_canvases, phapl_all_points)
{
var canvas_properties = {};
var all_points = phapl_all_points;
var i, j;
// draw phase planes
for (j in phapl_canvases) {
var cid = phapl_canvases[j][0];
var hl_fun = phapl_canvases[j][1];
var points = phapl_canvases[j][2];
var c = document.getElementById(cid);
var ctx = c.getContext('2d');
ctx.lineJoin = 'bevel';
var black_color = '#000000';
var red_color = '#ff0000';
var blue_color = '#0000ff';
var fg_color = '#8b008b';
var fg2_color = '#008000';
var height = c.height;
var single = (points.length == 1);
// Calculate borders
var pad = 1.0000001;
// var pad = 1.0;
// var pad = 2.0;
var min_x = points[0][0];
var max_x = points[0][0];
var min_y = points[0][1];
var max_y = points[0][1];
for (i in points) {
var x = points[i][0];
var y = points[i][1];
if (x > max_x) {
max_x = x;
}
if (y > max_y) {
max_y = y;
}
if (x < min_x) {
min_x = x
}
if (y < min_y) {
min_y = y;
}
}
min_x -= pad;
min_y -= pad;
max_x += pad;
max_y += pad;
// Expand to square
// ** After expansion assumption is used: max_x - min_x == max_y - min_y
var c_width = max_x - min_x;
var c_height = max_y - min_y;
var half_dif = Math.abs(c_width - c_height) / 2.0;
if (c_width > c_height) {
min_y -= half_dif;
max_y += half_dif;
} else {
min_x -= half_dif;
max_x += half_dif;
}
var mid_x = (min_x + max_x) / 2.0;
var mid_y = (min_y + max_y) / 2.0;
var dif_x = max_x - min_x;
var dif_y = max_y - min_y;
var height_dif_x = height / dif_x;
var is_linear_twin = false;
var nn = cid.replace('_nonlinear_', '_linear_');
if (nn === cid) {
nn = cid.replace('_linear_', '_nonlinear_');
if (nn != cid) {
is_linear_twin = true;
}
}
if (nn === cid) {
nn = false;
}
if (is_linear_twin) {
all_points = points;
} else {
all_points = phapl_all_points;
}
var line_width = 1.0 * dif_x / height;
canvas_properties[cid] = {
// function to draw half line
'hl_fun' : hl_fun,
'elem' : c,
'twin_cid' : nn,
'ctx' : ctx,
'mid_x' : mid_x,
'mid_y' : mid_y,
'min_x' : min_x,
'min_y' : min_y,
'max_x' : max_x,
'max_y' : max_y,
'dif_x' : dif_x,
'dif_y' : dif_y,
'height_dif_x' : height_dif_x,
'line_width' : line_width
};
// console.log(j, min_x, mid_x, max_x);
// console.log(j, min_y, mid_y, max_y);
// ctx.fillStyle = '#ffffff';
// ctx.fillStyle = '#ff0000';
// ctx.fillRect(0, 0, height, height);
// Scale and translate
ctx.scale(height / dif_x, height / dif_y);
ctx.translate(-min_x, -min_y);
// ctx.fillStyle = '#ffffff';
// ctx.fillRect(min_x, min_y, dif_x, dif_y);
var invert = function (y) { return mid_y - (y - mid_y); };
// var invert = function (y) { return y; };
for (j in all_points) {
var px = all_points[j][0];
var py = all_points[j][1];
// Circles around special points
ctx.strokeStyle = '#ff0000';
ctx.lineWidth = line_width;
ctx.beginPath();
ctx.arc(px, invert(py), 10.0 * dif_x / height, 0, 2 * Math.PI);
ctx.stroke();
}
var f = function (x, y) {
// console.log(x, y);
// ctx.fillStyle = '#ff0000';
// ctx.fillRect(x, mid_y - (y - mid_y), 0.1, 0.1);
var f2 = function (x, y, direction) {
hl_fun(ctx, direction, x, y, single,
black_color, fg_color, fg2_color,
// %% height * 5, height,
height * 2, height,
min_x, mid_x, max_x,
min_y, mid_y, max_y,
height_dif_x, line_width / 2
);
};
f2(x, y, -1);
f2(x, y, 1);
};
// Initial lines
for (j in points) {
var px = points[j][0];
var py = points[j][1];
// ctx.strokeStyle = '#ff0000';
// ctx.lineWidth = line_width / 2;
// ctx.strokeRect(px - 1, invert(py) - 1, 2, 2);
var ji;
// var parts = Math.floor(30 / points.length);
// // var parts = 30;
// for (ji = 0; ji < parts + 1; ji++) {
// f(px - 1 + ji * 2.0 / parts, py - 1);
// f(px - 1 + ji * 2.0 / parts, py + 1);
// f(px - 1, py - 1 + ji * 2.0 / parts);
// f(px + 1, py - 1 + ji * 2.0 / parts);
// }
}
// for (ji = 0; ji < 40 + 1; ji++) {
// f(Math.random() * dif_x + min_x, Math.random() * dif_y + min_y);
// }
var w = dif_x / 10.0;
for (var tx = min_x; tx <= max_x; tx += w) {
for (var ty = min_y; ty <= max_y; ty += w) {
f(tx + Math.random() * w, ty + Math.random() * w);
}
}
var shift = 10.0 * dif_x / height;
for (j in all_points) {
var px = all_points[j][0];
var py = all_points[j][1];
f(px - shift, py);
f(px, py - shift);
f(px + shift, py);
f(px, py + shift);
}
// Grid
ctx.lineWidth = line_width;
var line = function (ctx, x1, y1, x2, y2) {
ctx.beginPath();
ctx.moveTo(x1, y1);
ctx.lineTo(x2, y2);
ctx.stroke();
};
// We don't draw grid if there are too many lines.
// %% We might draw grid under trajectories when there are many lines but less than pixels in canvas.
if (max_x - min_x < 50.0) {
ctx.strokeStyle = 'lightgrey';
for (i = Math.floor(min_x) - 1; i < Math.floor(max_x) + 1; i += 1.0) {
if (i == 0.0) {
continue;
}
line(ctx, i, invert(min_y - 1), i, invert(max_y + 1));
}
for (i = Math.floor(min_y) - 1; i < Math.floor(max_y) + 1; i += 1.0) {
if (i == 0.0) {
continue;
}
line(ctx, min_x - 1, invert(i), max_x + 1, invert(i));
}
}
ctx.strokeStyle = 'violet';
line(ctx, 0.0, invert(min_y - 1), 0.0, invert(max_y + 1));
line(ctx, min_x - 1, invert(0.0), max_x + 1, invert(0.0));
// Triangles to mark axis and special points
var draw_triangle = function (x, y, color, orientation, size) {
ctx.fillStyle = color;
ctx.beginPath();
if (orientation) {
// vertical, like A
ctx.moveTo(x - size / 2, invert(y));
ctx.lineTo(x, invert(y + size));
ctx.lineTo(x + size / 2, invert(y));
} else {
// horizontal, like |>
ctx.moveTo(x, invert(y - size / 2));
ctx.lineTo(x + size, invert(y));
ctx.lineTo(x, invert(y + size / 2));
}
ctx.fill();
};
var axis_size = 15.0 * dif_x / height;
draw_triangle(0.0, min_y, black_color, true, axis_size);
draw_triangle(min_x, 0.0, black_color, false, axis_size);
var point_size = 10.0 * dif_x / height;
for (j in all_points) {
var px = all_points[j][0];
var py = all_points[j][1];
draw_triangle(px, min_y, red_color, true, point_size);
draw_triangle(min_x, py, red_color, false, point_size);
}
// Eigen vectors
for (j in all_points) {
if (all_points[j].length > 2) {
var px = all_points[j][0];
var py = all_points[j][1];
var vv = [ 0 ]
if (all_points[j].length > 4) {
vv.push(1);
}
for (var jj in vv) {
var vx = all_points[j][2 + 2 * jj];
var vy = all_points[j][3 + 2 * jj];
// normalize to 0.5 in length
var m = Math.sqrt(0.25 / (vx * vx + vy * vy));
vx *= m;
vy *= m;
ctx.strokeStyle = 'cyan';
line(ctx, px - vx, invert(py - vy), px + vx, invert(py + vy));
}
}
}
canvas_properties[cid].saved_canvas = ctx.getImageData(0, 0, c.width, c.height);
// Actions for mouse
// %% lift functions, don't create every time; is it important at all?
c.addEventListener('mouseout', function (e) {
var c = e.srcElement;
if (!c) {
c = e.originalTarget;
}
var d = canvas_properties[c.id];
d.ctx.putImageData(d.saved_canvas, 0, 0);
var nn = d.twin_cid;
if (nn != false) {
var d2 = canvas_properties[nn];
d2.ctx.putImageData(d2.saved_canvas, 0, 0);
}
});
c.onmousemove = (function (e) {
var c = e.srcElement;
if (!c) {
c = e.originalTarget;
}
var d = canvas_properties[c.id];
var rect = c.getBoundingClientRect();
var mx = e.clientX - rect.left;
var my = e.clientY - rect.top;
mx = mx / rect.width * d.dif_x + d.min_x;
my = d.mid_y - ((my / rect.height * d.dif_y + d.min_y) - d.mid_y);
d.ctx.putImageData(d.saved_canvas, 0, 0);
d.hl_fun(d.ctx, 1.0, mx, my, false,
red_color, fg_color, fg2_color,
// %% c.height * 3, c.height,
c.height * 1.5, c.height,
d.min_x, d.mid_x, d.max_x,
d.min_y, d.mid_y, d.max_y,
d.height_dif_x, d.line_width * 4
);
d.hl_fun(d.ctx, -1.0, mx, my, false,
blue_color, fg_color, fg2_color,
// %% c.height * 3, c.height,
c.height * 1.5, c.height,
d.min_x, d.mid_x, d.max_x,
d.min_y, d.mid_y, d.max_y,
d.height_dif_x, d.line_width * 4
);
var nn = d.twin_cid;
if (nn != false) {
var d2 = canvas_properties[nn];
mx = e.clientX - rect.left;
my = e.clientY - rect.top;
mx = mx / rect.width * d2.dif_x + d2.min_x;
my = d2.mid_y - ((my / rect.height * d2.dif_y + d2.min_y) - d2.mid_y);
d2.ctx.putImageData(d2.saved_canvas, 0, 0);
d2.hl_fun(d2.ctx, 1.0, mx, my, false,
red_color, fg_color, fg2_color,
// %% see similar above
d2.elem.height * 1.5, d2.elem.height,
d2.min_x, d2.mid_x, d2.max_x,
d2.min_y, d2.mid_y, d2.max_y,
d2.height_dif_x, d2.line_width * 4
);
d2.hl_fun(d2.ctx, -1.0, mx, my, false,
blue_color, fg_color, fg2_color,
// %% see similar above
d2.elem.height * 1.5, d2.elem.height,
d2.min_x, d2.mid_x, d2.max_x,
d2.min_y, d2.mid_y, d2.max_y,
d2.height_dif_x, d2.line_width * 4
);
}
});
}
}
phapl_populate_cache('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');
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Language / язык: <b>English</b>, <a href="phapl.ru.html">Русский</a>.
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<h2>PhaPl</h2>
<p>PhaPl is a software to research and plot phase portraits of autonomous systems of 2 differential equations on a plane. PhaPl works as a site or as a local html-page that works offline.</p>
<p><a href="#" onclick="scroll_to('phapl_div_choose'); return false">Go to task input/choice below.</a></p>
<p><a href="https://github.com/phapl/phapl.github.io/archive/master.zip">Download archive for local offline use</a> (~9 MB to download, ~37 MB to be stored unpacked on disk).</p>
<p>This version was published on: 2018-07-29 14:31 +0300.</p>
<p>"Math Processing Error" everywhere should be a problem with cache. Refreshing of page bypassing cache should help (press Shift-F5 in Chrome or Control-F5 in Firefox).</p>
<p>You are welcome to send your suggestions and reports about errors / bugs / problems through <a href="https://github.com/phapl/phapl.github.io/issues">GitHub</a>.</p>
<p>PhaPl uses modified SymPy, PyPy.js, MathJax, LZMA-JS. There is full info about that in <a href="https://github.com/phapl/phapl">the repo with the sources of PhaPl</a>.</p>
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<p>PhaPl wants to be Free Software, but this page can include sets of tasks that prohibit redistribution.</p>
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<input type="button" id="block1-header" onclick="collapse('block1')" value="► Show tasks: Tasks 16 (main.pdf)">
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Copyright © 2010 Astashova I.V., Nikishkin V.A.<br><i>Astashova I.V., Nikishkin V.A.</i> Practicum on course "Differential equations". Tutorial. 3rd edition, revised. Moscow: Publishing Center of EOI, 2010. 94 p., illustrated. URL: <a href="http://new.math.msu.su/diffur/main_du_2010.pdf">http://new.math.msu.su/diffur/main_du_2010.pdf</a></br>
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<td onclick="phapl_set(this, '3 * x - y', '13 * x - 3 * y')">1. \( \left\{ \begin{aligned}\dot{x} &= 3 x - y \\\dot{y} &= 13 x - 3 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x - 3 * y', '5 * x + 9 * y')">11. \( \left\{ \begin{aligned}\dot{x} &= x - 3 y \\\dot{y} &= 5 x + 9 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '7 * x - 4 * y', '4 * x - y')">21. \( \left\{ \begin{aligned}\dot{x} &= 7 x - 4 y \\\dot{y} &= 4 x - y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x - 3 * y', '7 * x - 9 * y')">2. \( \left\{ \begin{aligned}\dot{x} &= x - 3 y \\\dot{y} &= 7 x - 9 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x - y', '4 * x - 3 * y')">12. \( \left\{ \begin{aligned}\dot{x} &= 2 x - y \\\dot{y} &= 4 x - 3 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x - y', 'x + y')">22. \( \left\{ \begin{aligned}\dot{x} &= x - y \\\dot{y} &= x + y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '5 * x - 3 * y', '3 * x - y')">3. \( \left\{ \begin{aligned}\dot{x} &= 5 x - 3 y \\\dot{y} &= 3 x - y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '4 * x - 5 * y', '5 * x - 4 * y')">13. \( \left\{ \begin{aligned}\dot{x} &= 4 x - 5 y \\\dot{y} &= 5 x - 4 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '3 * x + 2 * y', '3 * x + 4 * y')">23. \( \left\{ \begin{aligned}\dot{x} &= 3 x + 2 y \\\dot{y} &= 3 x + 4 y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x - 9 * y', 'x + y')">4. \( \left\{ \begin{aligned}\dot{x} &= x - 9 y \\\dot{y} &= x + y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '-7 * x + 3 * y', '-x - 3 * y')">14. \( \left\{ \begin{aligned}\dot{x} &= - 7 x + 3 y \\\dot{y} &= - x - 3 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '3 * x - 4 * y', 'x - 2 * y')">24. \( \left\{ \begin{aligned}\dot{x} &= 3 x - 4 y \\\dot{y} &= x - 2 y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '4 * x + y', '2 * x + 5 * y')">5. \( \left\{ \begin{aligned}\dot{x} &= 4 x + y \\\dot{y} &= 2 x + 5 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '9 * x - 5 * y', '5 * x - y')">15. \( \left\{ \begin{aligned}\dot{x} &= 9 x - 5 y \\\dot{y} &= 5 x - y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x - y', '5 * x - 2 * y')">25. \( \left\{ \begin{aligned}\dot{x} &= 2 x - y \\\dot{y} &= 5 x - 2 y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '5 * x - 3 * y', '4 * x - 3 * y')">6. \( \left\{ \begin{aligned}\dot{x} &= 5 x - 3 y \\\dot{y} &= 4 x - 3 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x - 2 * y', '2 * x + y')">16. \( \left\{ \begin{aligned}\dot{x} &= x - 2 y \\\dot{y} &= 2 x + y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x - 2 * y', '7 * x - 8 * y')">26. \( \left\{ \begin{aligned}\dot{x} &= x - 2 y \\\dot{y} &= 7 x - 8 y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x - 2 * y', '13 * x - y')">7. \( \left\{ \begin{aligned}\dot{x} &= x - 2 y \\\dot{y} &= 13 x - y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '8 * x - 3 * y', '2 * x + 3 * y')">17. \( \left\{ \begin{aligned}\dot{x} &= 8 x - 3 y \\\dot{y} &= 2 x + 3 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '3 * x - 2 * y', '2 * x - y')">27. \( \left\{ \begin{aligned}\dot{x} &= 3 x - 2 y \\\dot{y} &= 2 x - y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '-5 * x + y', '2 * x - 4 * y')">8. \( \left\{ \begin{aligned}\dot{x} &= - 5 x + y \\\dot{y} &= 2 x - 4 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x + y', '4 * x - y')">18. \( \left\{ \begin{aligned}\dot{x} &= 2 x + y \\\dot{y} &= 4 x - y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '4 * x - 2 * y', 'x + 2 * y')">28. \( \left\{ \begin{aligned}\dot{x} &= 4 x - 2 y \\\dot{y} &= x + 2 y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '6 * x - y', 'x + 4 * y')">9. \( \left\{ \begin{aligned}\dot{x} &= 6 x - y \\\dot{y} &= x + 4 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x - 5 * y', '4 * x - 2 * y')">19. \( \left\{ \begin{aligned}\dot{x} &= 2 x - 5 y \\\dot{y} &= 4 x - 2 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '5 * x + 3 * y', 'x + 3 * y')">29. \( \left\{ \begin{aligned}\dot{x} &= 5 x + 3 y \\\dot{y} &= x + 3 y \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x - 2 * y', 'x + 3 * y')">10. \( \left\{ \begin{aligned}\dot{x} &= x - 2 y \\\dot{y} &= x + 3 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '-2 * x - 3 * y', '4 * x - 9 * y')">20. \( \left\{ \begin{aligned}\dot{x} &= - 2 x - 3 y \\\dot{y} &= 4 x - 9 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x - y', '5 * x - 4 * y')">30. \( \left\{ \begin{aligned}\dot{x} &= 2 x - y \\\dot{y} &= 5 x - 4 y \end{aligned}\right. \)</td>
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</table>
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<form>
<input type="button" id="block2-header" onclick="collapse('block2')" value="► Show tasks: Tasks 17 (main.pdf)">
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Copyright © 2010 Astashova I.V., Nikishkin V.A.<br><i>Astashova I.V., Nikishkin V.A.</i> Practicum on course "Differential equations". Tutorial. 3rd edition, revised. Moscow: Publishing Center of EOI, 2010. 94 p., illustrated. URL: <a href="http://new.math.msu.su/diffur/main_du_2010.pdf">http://new.math.msu.su/diffur/main_du_2010.pdf</a></br>
<b>Comments:</b> Task 22 got additional parentheses for ln(). The equilibrium points are not convenient otherwise.</br>
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<td onclick="phapl_set(this, '(x + y)^2 - 1', '- y^2 - x + 1')">1. \( \left\{ \begin{aligned}\dot{x} &= \left(x + y\right)^{2} - 1 \\\dot{y} &= - x - y^{2} + 1 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'sqrt(x^2 - y + 2) - 2', 'arctg(x^2 + x * y)')">11. \( \left\{ \begin{aligned}\dot{x} &= \sqrt{x^{2} - y + 2} - 2 \\\dot{y} &= \operatorname{arctg}{\left (x \left(x + y\right) \right )} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x * y - 4', '(x - 4) * (y - x)')">21. \( \left\{ \begin{aligned}\dot{x} &= x y - 4 \\\dot{y} &= - \left(x - 4\right) \left(x - y\right) \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '- 2 * y - x^3', '3 * x - 4 * y^3')">2. \( \left\{ \begin{aligned}\dot{x} &= - x^{3} - 2 y \\\dot{y} &= 3 x - 4 y^{3} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'sqrt((x - y)^2 + 3) - 2', 'e^(y^2 - x) - e')">12. \( \left\{ \begin{aligned}\dot{x} &= \sqrt{\left(x - y\right)^{2} + 3} - 2 \\\dot{y} &= - e + e^{- x + y^{2}} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x^2 - y', 'ln(1 - x + x^2) - ln(3)')">22. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - y \\\dot{y} &= \log{\left (x^{2} - x + 1 \right )} - \log{\left (3 \right )} \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x + y + 1', 'y + sqrt(1 + 2 * x^2)')">3. \( \left\{ \begin{aligned}\dot{x} &= x + y + 1 \\\dot{y} &= y + \sqrt{2 x^{2} + 1} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '3 - sqrt(4 - x^2 + y)', 'ln(x^2 -3)')">13. \( \left\{ \begin{aligned}\dot{x} &= - \sqrt{- x^{2} + y + 4} + 3 \\\dot{y} &= \log{\left (x^{2} - 3 \right )} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'y', 'sin(x + y)')">23. \( \left\{ \begin{aligned}\dot{x} &= y \\\dot{y} &= \sin{\left (x + y \right )} \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x^2 - y', '(x - y) * (x - y + 2)')">4. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - y \\\dot{y} &= \left(x - y\right) \left(x - y + 2\right) \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x + y^2 - 1', '6 * x - y^2 + 1')">14. \( \left\{ \begin{aligned}\dot{x} &= 2 x + y^{2} - 1 \\\dot{y} &= 6 x - y^{2} + 1 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'ln(y^2 - x)', 'x - y - 1')">24. \( \left\{ \begin{aligned}\dot{x} &= \log{\left (- x + y^{2} \right )} \\\dot{y} &= x - y - 1 \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'ln((y^2 - y + 1) / 3)', 'x^2 - y^2')">5. \( \left\{ \begin{aligned}\dot{x} &= \log{\left (\frac{y^{2}}{3} - \frac{y}{3} + \frac{1}{3} \right )} \\\dot{y} &= x^{2} - y^{2} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '(2 * x - y)^2 - 9', '9 - (x - 2 * y)^2')">15. \( \left\{ \begin{aligned}\dot{x} &= \left(2 x - y\right)^{2} - 9 \\\dot{y} &= - \left(x - 2 y\right)^{2} + 9 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '(x - y) * (x + 3)', 'y - 1')">25. \( \left\{ \begin{aligned}\dot{x} &= \left(x + 3\right) \left(x - y\right) \\\dot{y} &= y - 1 \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '2 * x * y - 4 * y - 8', '4 * y^2 - x^2')">6. \( \left\{ \begin{aligned}\dot{x} &= 2 x y - 4 y - 8 \\\dot{y} &= - x^{2} + 4 y^{2} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'e^y - e^x', 'sqrt(3 * x + y^2) - 2')">16. \( \left\{ \begin{aligned}\dot{x} &= - e^{x} + e^{y} \\\dot{y} &= \sqrt{3 x + y^{2}} - 2 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'y^2 - 1', '(x - 1) * (x - 2 * y)')">26. \( \left\{ \begin{aligned}\dot{x} &= y^{2} - 1 \\\dot{y} &= \left(x - 1\right) \left(x - 2 y\right) \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '4 - 4 * x - 2 * y', 'x * y')">7. \( \left\{ \begin{aligned}\dot{x} &= - 4 x - 2 y + 4 \\\dot{y} &= x y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '2 * x + y^2 - 1', '6 * x - y^2 + 1')">17. \( \left\{ \begin{aligned}\dot{x} &= 2 x + y^{2} - 1 \\\dot{y} &= 6 x - y^{2} + 1 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '(y - 1) * (y - x)', 'x^2 + 3 * y + 2')">27. \( \left\{ \begin{aligned}\dot{x} &= - \left(x - y\right) \left(y - 1\right) \\\dot{y} &= x^{2} + 3 y + 2 \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'y^2 - 4 * x^2', '4 * y - 8')">8. \( \left\{ \begin{aligned}\dot{x} &= - 4 x^{2} + y^{2} \\\dot{y} &= 4 y - 8 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x^2 - y^2 - 1', '2 * y')">18. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - y^{2} - 1 \\\dot{y} &= 2 y \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x * y - 4 * y - 3 * x^2 + 12 * x', 'x^2 - 1')">28. \( \left\{ \begin{aligned}\dot{x} &= - 3 x^{2} + x y + 12 x - 4 y \\\dot{y} &= x^{2} - 1 \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, '(2 * x - y) * (x - 3)', 'x * y - 3')">9. \( \left\{ \begin{aligned}\dot{x} &= \left(x - 3\right) \left(2 x - y\right) \\\dot{y} &= x y - 3 \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '1 - x^2 - y^2', '2 * x')">19. \( \left\{ \begin{aligned}\dot{x} &= - x^{2} - y^{2} + 1 \\\dot{y} &= 2 x \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x * y - y - 2 * x^2 + 2 * x', '(y - 1) * (x + 3)')">29. \( \left\{ \begin{aligned}\dot{x} &= - 2 x^{2} + x y + 2 x - y \\\dot{y} &= \left(x + 3\right) \left(y - 1\right) \end{aligned}\right. \)</td>
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<td onclick="phapl_set(this, 'x^2 + y^2 - 6 * x - 8 * y', 'x * (2 * y - x + 5)')">10. \( \left\{ \begin{aligned}\dot{x} &= x^{2} - 6 x + y^{2} - 8 y \\\dot{y} &= x \left(- x + 2 y + 5\right) \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, 'x^2 + y', 'x^2 - (y + 1)^2')">20. \( \left\{ \begin{aligned}\dot{x} &= x^{2} + y \\\dot{y} &= x^{2} - \left(y + 1\right)^{2} \end{aligned}\right. \)</td>
<td onclick="phapl_set(this, '(x + y) * (y - 2)', 'x - 1')">30. \( \left\{ \begin{aligned}\dot{x} &= \left(x + y\right) \left(y - 2\right) \\\dot{y} &= x - 1 \end{aligned}\right. \)</td>
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<!-- %% move into code generator -->
<a href="#" onclick="phapl_gen_linear('1', true); return false">Unstable node</a><br>
<a href="#" onclick="phapl_gen_linear('2', true); return false">Stable node</a><br>
<a href="#" onclick="phapl_gen_linear('3', true); return false">Saddle</a><br>
<a href="#" onclick="phapl_gen_linear('4', true); return false">Centre</a><br>
<a href="#" onclick="phapl_gen_linear('5', true); return false">Unstable focus</a><br>
<a href="#" onclick="phapl_gen_linear('6', true); return false">Stable focus</a><br>
<a href="#" onclick="phapl_gen_linear('7', true); return false">Unstable degenerate node</a><br>
<a href="#" onclick="phapl_gen_linear('7a', true); return false">Unstable dicritical node</a><br>
<a href="#" onclick="phapl_gen_linear('8', true); return false">Stable degenerate node</a><br>
<a href="#" onclick="phapl_gen_linear('8a', true); return false">Stable dicritical node</a><br>
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