Systems of differential equations

Display solutions of the ordinary differential equation

[math]\displaystyle{ y_1'= f_1(x,y_1,y_2) }[/math]

[math]\displaystyle{ y_2'= f_2(x,y_1,y_2) }[/math]

with initial values [math]\displaystyle{ (x_0,c_1) }[/math], [math]\displaystyle{ (x_0,c_2) }[/math].

f₁(x,y1,y2)=
f₂(x,y1,y2)=

See also

• Differential equations
• Lotka-Volterra equations
• Epidemiology: The SIR model
• Population growth models
• Autocatalytic process
• Logistic process

The underlying JavaScript code

<form>
f<sub>1</sub>(x,y1,y2)=<input type="text" id="odeinput1" value="y1+y2"><br />
f<sub>2</sub>(x,y1,y2)=<input type="text" id="odeinput2" value="y2+1"><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 P1 = brd.create('point',[0,1], {name:'(x_0,c_1)'});
var line = brd.create('line',[function(){return -P1.X();},function(){return 1;},function(){return 0;}],{visible:false});
var P2 = brd.create('glider',[0,2,line], {name:'(x_0,c_2)'});

function doIt() {
  var txt1 = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput1").value);
  var txt2 = JXG.GeonextParser.geonext2JS(document.getElementById("odeinput2").value);
  f = new Function("x", "yy", "var y1 = yy[0], y2 = yy[1];  var z1 = " + txt1 + "; var z2 = " + txt2 + "; return [z1,z2];");
  brd.update();
}

function ode() {
   return JXG.Math.Numerics.rungeKutta(JXG.Math.Numerics.predefinedButcher.Heun, [P1.Y(),P2.Y()], [P1.X(), P1.X()+N.Value()], 200, f);
}

var g1 = brd.createElement('curve', [[0],[0]], {strokeColor:'red', strokeWidth:'2px', name:'y_1', withLabel:false});
var g2 = brd.createElement('curve', [[0],[0]], {strokeColor:'black', strokeWidth:'2px', name:'y_2', withLabel:false});
g1.updateDataArray = function() {
    var data = ode();
    var h = N.Value()/200;
    this.dataX = [];
    this.dataY = [];
    for(var i=0; i<data.length; i++) {
        this.dataX[i] = P1.X()+i*h;
        this.dataY[i] = data[i][0];
    }
};
g2.updateDataArray = function() {
    var data = ode();
    var h = N.Value()/200;
    this.dataX = [];
    this.dataY = [];
    for(var i=0; i<data.length; i++) {
        this.dataX[i] = P2.X()+i*h;
        this.dataY[i] = data[i][1];
    }
};
doIt();