differential equations
system:sage


{{{id=1|
# slightly modified from the interact webpage

def picard_iteration(f, t0, u0, iterations):
    '''
    Computes the N-th Picard iterate for the IVP  

        u'(t) = f(t,u(t)), for all t, u(t0) = u0.

    '''
    if iterations == 0:
        u1 = lambda t: u0 + 2*t-t^2
        return expand(u1(t))
    for i in range(iterations):
        u_old = lambda s: picard_iteration(f, t0, u0, iterations-1).subs(t=s)
        un = lambda t: u0 + integral(f(t=s,u=u_old(s)), s, t0, t)
    return expand(un(t))

@interact
def picarder(n_iterations = slider(0,20,1,default = 2)):
    var('y,u,t,s')
    f = lambda t,u: u
    y = function('y',t)   # define x to be a function of that variable
    DE = diff(y, t) - f(t,y)
    g=desolve(DE,[y,t],[0,1])
    exact = plot(g,(t,0,2))
    html('The exact solution is:<br>')
    html('$'+latex(g)+'$')
    for i in range(n_iterations):
        pic = picard_iteration(f,0,1,i)
        html('The Picard iteration approximation after ' + str(i) + ' iterations is:<br>')
        html('$'+latex(pic)+'$')
        exact = exact+plot(pic,(t,0,2), rgbcolor = (1,0,0))
    show(exact)
///
<html><!--notruncate-->
        <div padding=6 id="div-interact-1">
          <table width=800px height=20px bgcolor="#c5c5c5" cellpadding=15>
            <tr>
              <td bgcolor="#f9f9f9" valign=top align=left>
            <table>
              <tr><td colspan=3><table><tr><td align=right><font color="black">n_iterations&nbsp;</font></td><td>
        <table>
          <tr>
            <td>
              <div id="slider-n_iterations-1" style="margin:0px; margin-left: 1.0em; margin-right: 1.0em; width: 15.0em;"></div>
            </td>
            
            <td>
              <font color="black" id="slider-n_iterations-1-lbl"></font>
            </td>
          </tr>
        </table><script>
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                value: 2,
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                        $('#slider-n_iterations-1-lbl').text(values[position]);
                        interact(1, {variable: 'n_iterations', adapt_number: 1, value: encode64(position)}, 1);
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</tr></table></td></tr>
              <tr><td></td><td style='width: 100%;'>
        <div id="cell-interact-1"><?__SAGE__START>
          <table border=0 bgcolor="white" width=100%>
            <tr>
              <td bgcolor="white" align=left valign=top>
                <pre><?__SAGE__TEXT></pre>
              </td>
            </tr>
            <tr>
              <td align=left valign=top><?__SAGE__HTML></td>
            </tr>
          </table><?__SAGE__END>
        </div></td><td></td></tr>
              <tr><td colspan=3></td></tr>
            </table></td>
            </tr>
          </table>
        </div></html>
}}}

{{{id=5|
# slightly modified from the interact webpage

x,y = var('x,y')
from sage.ext.fast_eval import fast_float
@interact
def _(f = input_box(default=1), g=input_box(default=y^2),
      xmin=input_box(default=-3), xmax=input_box(default=3),
      ymin=input_box(default=-3), ymax=input_box(default=3),
      start_x=input_box(default=0.5), start_y=input_box(default=0.5),
      step_size=(0.01,(0.001, 0.2)), steps=(600,(0, 1400)) ):
    ff = fast_float(f, 'x', 'y')
    gg = fast_float(g, 'x', 'y')
    steps = int(steps)

    points = [ (start_x, start_y) ]
    for i in range(steps):
        xx, yy = points[-1]
        try:
            points.append( (xx + step_size * ff(xx,yy), yy + step_size * gg(xx,yy)) )
        except (ValueError, ArithmeticError, TypeError):
            break

    starting_point = point(points[0], pointsize=50)
    solution = line(points)
    vector_field = plot_vector_field( (f,g), (x,xmin,xmax), (y,ymin,ymax) )

    result = vector_field + starting_point + solution

    html(r"$\displaystyle\frac{dx}{dt} = %s$  $ \displaystyle\frac{dy}{dt} = %s$" % (latex(f),latex(g)))
    result.show(xmin=xmin,xmax=xmax,ymin=ymin,ymax=ymax)
///
<html><!--notruncate-->
        <div padding=6 id="div-interact-5">
          <table width=800px height=20px bgcolor="#c5c5c5" cellpadding=15>
            <tr>
              <td bgcolor="#f9f9f9" valign=top align=left>
            <table>
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</tr><tr><td align=right><font color="black">g&nbsp;</font></td><td><input type="text" value="y^2" size=80 onchange="interact(5, {variable: 'g', adapt_number: 33, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">xmin&nbsp;</font></td><td><input type="text" value="-3" size=80 onchange="interact(5, {variable: 'xmin', adapt_number: 34, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">xmax&nbsp;</font></td><td><input type="text" value="3" size=80 onchange="interact(5, {variable: 'xmax', adapt_number: 35, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">ymin&nbsp;</font></td><td><input type="text" value="-3" size=80 onchange="interact(5, {variable: 'ymin', adapt_number: 36, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">ymax&nbsp;</font></td><td><input type="text" value="3" size=80 onchange="interact(5, {variable: 'ymax', adapt_number: 37, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">start_x&nbsp;</font></td><td><input type="text" value="0.500000000000000" size=80 onchange="interact(5, {variable: 'start_x', adapt_number: 38, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">start_y&nbsp;</font></td><td><input type="text" value="0.500000000000000" size=80 onchange="interact(5, {variable: 'start_y', adapt_number: 39, value: encode64(this.value)}, 1)"></input></td>
</tr><tr><td align=right><font color="black">step_size&nbsp;</font></td><td>
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            <td>
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</tr><tr><td align=right><font color="black">steps&nbsp;</font></td><td>
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