{{{id=29|
# Copyright 2009, P. Lutus
# Released under the GPL (http://www.gnu.org/copyleft/gpl.html)
///
}}}
{{{id=3|
%auto
# reset() # commented out for now -- ticket 7255
# special equation rendering
def render(x,name = "temp.png",size = "normal"):
if(type(x) != type("")): x = latex(x)
latex.eval("\\" + size + " $" + x + "$",{},"",name)
var('a b r c c_t x y z t q k1 k2 k_1 k_2 omega')
def fa(t):
a = [0,1,0,-1,0]
return a[int(t)]
///
}}}
{{{id=30|
render(y == x^2,"this.png","Large")
///
}}}
{{{id=8|
# example chart code block
dt = 100.0
pos = []
vel = []
acc = []
p = 0.0
v = 0.0
a = 0.0
ppos = 0
vpos = 2.5
apos = 5
pcolor = 'blue'
vcolor = '#008000'
acolor = '#800000'
for st in range(5 * dt):
t = st/dt
a = fa(t)
v += a / dt
p += v / dt
pos.append(p+ppos)
vel.append(v+vpos)
acc.append(a+apos)
alp = list_plot(acc,plotjoined=True,rgbcolor=acolor)
vlp = list_plot(vel,plotjoined=True,rgbcolor=vcolor)
posp = list_plot(pos,plotjoined=True,rgbcolor=pcolor)
lines = Graphics()
labels = Graphics()
for n in range(0,6):
lines += line([(n * dt,0),(n*dt,6.3)],linestyle='--',rgbcolor='#808080')
labels += text("%.0f sec" % (n*20),(n*dt,6.6),rgbcolor='black')
for x in range(0,2):
xm = x*dt*5.4-20
for n in range(-1,2):
labels += text("%+d" % n,(xm,apos+n),rgbcolor=acolor)
for n in range(0,2):
labels += text("%+d" % n,(xm,vpos+n),rgbcolor=vcolor)
for n in range(0,3):
labels += text("%+d" % n,(xm,ppos+n),rgbcolor=pcolor)
show((alp+vlp+posp+lines+labels),axes=False)
///
}}}
{{{id=9|
render(latex("$y'(t) = \int y''(t) \, dt + C$"),"diffeq_acc_vel_integral.png","large")
///
}}}
{{{id=10|
render(latex("$y(t) = \int y'(t) \, dt + C$"),"diffeq_vel_pos_integral.png","large")
///
}}}
{{{id=11|
render(latex("$y'(t) = \\frac{d}{dt} y(t)$"),"diffeq_vel_pos_deriv.png","large")
///
}}}
{{{id=12|
render(latex("$y''(t) = \\frac{d}{dt} y'(t)$"),"diffeq_acc_vel_deriv.png","large")
///
}}}
{{{id=13|
render(latex("$y(t) = \\frac{d}{dt} (\\int y(t) \, dt)$"),"diffeq_fundamental_theorem.png","large")
///
}}}
{{{id=19|
render(latex("$y(t) + r c y'(t) = 0$"),"diffeq_first_example1.png","large")
///
}}}
{{{id=20|
render(latex("$y(0) = 1$"),"diffeq_first_example2.png","large")
///
}}}
{{{id=14|
var('y,r,c,t')
y = function('y',t)
de = y + r*c*diff(y,t) == 0
des = desolve(de,[y,t],[0,1])
render(des,"diffeq_first_example_solution.png","Large")
///
}}}
{{{id=21|
show(des)
///
e^{-\frac{t}{c r}}
}}}
{{{id=23|
f(t,r,c) = des
plot(f(t,1,1),(t,0,5),figsize=(4,3))
///
}}}
{{{id=15|
f(t,a,b,r,c) = (a-b) * des + b
render("$y(t) = " + latex(f(t,a,b,r,c)) + "$","diffeq_first_example_function.png","large")
///
}}}
{{{id=16|
# r = 10000 ohms, c = 100e-6 farads
lbl = text("$RC \, Circuit, \, r = 10000\Omega, \, c = 100\mu f $",(3,3),fontsize=12,rgbcolor='#006000')
p = plot(f(t,0,10,10000,100e-6),(t,0,5), figsize=(4,3),axes_labels=['time','v'])
show(p+lbl)
///
}}}
{{{id=24|
render(latex("$y(t) + r c y'(t) = sin(\\omega t)$"),"diffeq_second_example1.png","large")
///
}}}
{{{id=25|
y = function('y',t)
de = y + r*c_t*diff(y,t) == sin(omega * t)
des = desolve(de,[y,t])
render(des,"diffeq_second_example2.png","Large")
///
}}}
{{{id=26|
des2 = (des.subs(c == 0)).full_simplify()
df(r,c_t,omega,t) = des2
render(y == df(r,c,omega,t),"diffeq_second_example3.png","Large")
///
}}}
{{{id=27|
# r = 1000 ohms, c = 0.5e-6 farads, f = 440
r = 1000
c = 0.5e-6
f = 440
omega = 2*pi*f
pa = plot(sin(omega*t),(t,0,.005),rgbcolor=('#800000'),axes_labels=['time','v'])
pb = plot(df(r,c,omega,t),(t,0,.005),rgbcolor=('#008000'))
lbl = text("$f = %.0f Hz \,r = %.0f \Omega \, c = %.1f {\mu}f$"
% (f,r,c*1e6),(.0025,1.2),fontsize=12,rgbcolor='black')
show(pa+pb+lbl,figsize=(4,3))
///
}}}
{{{id=28|
///
}}}