Differential equations: Difference between revisions

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Display solutions of the ordinary differential equation
Display solutions of the ordinary differential equation
:<math> y'= f(x,y)</math>
:<math> y'= f(t,y)</math>
with initial value <math>(x_0,y_0)</math>.
with initial value <math>(t_0,y_0)</math>.
 
It is easy to incorporate sliders: give the slider a (unique) name and use this name in the equation. In the example below, the slider name is <math>c</math>.
<html>
<html>
<form>
<form>
f(x,y)=<input type="text" id="odeinput" value="(2-x)*y"><input type=button value="ok" onclick="doIt()">
f(t,y)=<input type="text" id="odeinput" value="(2-t)*y + c"><input type=button value="ok" onclick="doIt()">
</form>
</form>
</html>
</html>
Line 10: Line 12:
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var P = brd.create('point',[0,1], {name:'(x_0,y_0)'});
var slider = brd.create('slider',[[-7,8],[7,8],[-15,0,15]], {name:'c'});
var P = brd.create('point',[0,1], {name:'(t_0, y_0)'});
var f;


function doIt() {
function doIt() {
   var txt = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput").value);
   var snip = brd.jc.snippet(document.getElementById("odeinput").value, true, 't, y');
   f = new Function("x", "yy", "var y = yy[0]; var z = " + txt + "; return [z];");
   f = function (t, yy) {
      return [snip(t, yy[0])];
  }
   brd.update();
   brd.update();
}
}


function ode() {
function ode() {
   return JXG.Math.Numerics.rungeKutta(JXG.Math.Numerics.predefinedButcher.Heun, [P.Y()], [P.X(), P.X()+N.Value()], 200, f);
   return JXG.Math.Numerics.rungeKutta('heun', [P.Y()], [P.X(), P.X()+N.Value()], 200, f);
}
}


var g = brd.createElement('curve', [[0],[0]], {strokeColor:'red', strokeWidth:'2px'});
var g = brd.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2});
g.updateDataArray = function() {
g.updateDataArray = function() {
     var data = ode();
     var data = ode();
     var h = N.Value()/200;
     var h = N.Value()/200;
    var i;
     this.dataX = [];
     this.dataX = [];
     this.dataY = [];
     this.dataY = [];
     for(var i=0; i<data.length; i++) {
     for(i=0; i<data.length; i++) {
         this.dataX[i] = P.X()+i*h;
         this.dataX[i] = P.X()+i*h;
         this.dataY[i] = data[i][0];
         this.dataY[i] = data[i][0];
Line 37: Line 44:


===See also===
===See also===
* [[Systems of differential equations]]
* [[Lotka-Volterra equations]]
* [[Lotka-Volterra equations]]
* [[Epidemiology: The SIR model]]
* [[Epidemiology: The SIR model]]
Line 46: Line 54:
<source lang="xml">
<source lang="xml">
<form>
<form>
f(x,y)=<input type="text" id="odeinput" value="(2-x)*y"><input type=button value="ok" onclick="doIt()">
f(t,y)=<input type="text" id="odeinput" value="(2-t)*y + c"><input type=button value="ok" onclick="doIt()">
</form>
</form>
</source>
</source>
Line 52: Line 60:
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var P = brd.create('point',[0,1], {name:'(x_0,y_0)'});
var slider = brd.create('slider',[[-7,8],[7,8],[-15,0,15]], {name:'c'});
var P = brd.create('point',[0,1], {name:'(t_0, y_0)'});
var f;


function doIt() {
function doIt() {
   var txt = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput").value);
   var snip = brd.jc.snippet(document.getElementById("odeinput").value, true, 't, y');
   f = new Function("x", "yy", "var y = yy[0]; var z = " + txt + "; return [z]");
   f = function (t, yy) {
      return [snip(t, yy[0])];
  }
   brd.update();
   brd.update();
}
}


function ode() {
function ode() {
   return JXG.Math.Numerics.rungeKutta(JXG.Math.Numerics.predefinedButcher.Heun, [P.Y()], [P.X(), P.X()+N.Value()], 200, f);
   return JXG.Math.Numerics.rungeKutta('heun', [P.Y()], [P.X(), P.X()+N.Value()], 200, f);
}
}


var g = brd.createElement('curve', [[0],[0]], {strokeColor:'red', strokeWidth:'2px'});
var g = brd.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2});
g.updateDataArray = function() {
g.updateDataArray = function() {
     var data = ode();
     var data = ode();
     var h = N.Value()/200;
     var h = N.Value()/200;
    var i;
     this.dataX = [];
     this.dataX = [];
     this.dataY = [];
     this.dataY = [];
     for(var i=0; i<data.length; i++) {
     for(i=0; i<data.length; i++) {
         this.dataX[i] = P.X()+i*h;
         this.dataX[i] = P.X()+i*h;
         this.dataY[i] = data[i][0];
         this.dataY[i] = data[i][0];

Latest revision as of 08:46, 18 December 2020

Display solutions of the ordinary differential equation

[math]\displaystyle{ y'= f(t,y) }[/math]

with initial value [math]\displaystyle{ (t_0,y_0) }[/math].

It is easy to incorporate sliders: give the slider a (unique) name and use this name in the equation. In the example below, the slider name is [math]\displaystyle{ c }[/math].

f(t,y)=

See also

The underlying JavaScript code

<form>
f(t,y)=<input type="text" id="odeinput" value="(2-t)*y + c"><input type=button value="ok" onclick="doIt()">
</form>
var brd = JXG.JSXGraph.initBoard('jxgbox', {axis:true, boundingbox:[-11,11,11,-11]});
var N = brd.create('slider',[[-7,9.5],[7,9.5],[-15,10,15]], {name:'N'});
var slider = brd.create('slider',[[-7,8],[7,8],[-15,0,15]], {name:'c'});
var P = brd.create('point',[0,1], {name:'(t_0, y_0)'});
var f;

function doIt() {
  var snip = brd.jc.snippet(document.getElementById("odeinput").value, true, 't, y');
  f = function (t, yy) {
      return [snip(t, yy[0])];
  }
  brd.update();
}

function ode() {
   return JXG.Math.Numerics.rungeKutta('heun', [P.Y()], [P.X(), P.X()+N.Value()], 200, f);
}

var g = brd.create('curve', [[0],[0]], {strokeColor:'red', strokeWidth:2});
g.updateDataArray = function() {
    var data = ode();
    var h = N.Value()/200;
    var i;
    this.dataX = [];
    this.dataY = [];
    for(i=0; i<data.length; i++) {
        this.dataX[i] = P.X()+i*h;
        this.dataY[i] = data[i][0];
    }
};
doIt();