Differential equations

Display solutions of the ordinary differential equation

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

with initial value [math]\displaystyle{ (x_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(x,y)=

See also

• Systems of differential equations
• Lotka-Volterra equations
• Epidemiology: The SIR model
• Population growth models
• Autocatalytic process
• Logistic process
• Paul Pearson has written a very nice variation: Slope fields and solution curves (using the Runge-Kutta)

The underlying JavaScript code

<form>
f(x,y)=<input type="text" id="odeinput" value="(2-x)*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:'(x_0, y_0)'});
var f;

function doIt() {
  var snip = brd.jc.snippet(document.getElementById("odeinput").value, true, 'x, y');
  f = function (x, yy) {
      return [snip(x, 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();