Solving Separable Differential Equations
John Travis
Mississippi College
Blanchard, Devaney and Hall, 4th edition.
Once again, we consider various differential equations in the form
$\frac{\mathrm{dy} }{\mathrm{dt}} = f(t,y)$
Qualitative analysis can be appropriate when:
- it may be more difficult to discern an analytic approach to solving the equation
- all you want is an idea about how solutions behave over time
- you need only asymptotic behavior (as $t \to \infty$)
# Worksheet still under development. Use at your own risk! 8-o
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t,y,yy = var('t y yy')
def f(t,y):
# enter the desired right hand side below
return t^2+y
def ff(t,yy):
# enter the desired right hand side below
return t^2+yy
html('Using $f(t,y) = '+str(f(t,y))+'$')
yy = function('yy',t)
DE = diff(yy,t)== f(t,yy)
html('Differential Equation is $%s$'%str(latex(DE)))
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Using
Differential Equation is
Using
Differential Equation is |
tmin=-2
tmax=2
ymin=-2
ymax=2
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plot_slope_field(f(t,y),(t,tmin,tmax),(y,ymin,ymax)).show(aspect_ratio=1)
html('<center>Slope field for $dy/dt = %s$</center>'%str(f(t,y)))
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# Overlay particular slopes on the slope field above at an interactively chosen initial value. Also draw soln curve.
@interact
def _(t0 = slider(tmin,tmax,1/10,tmin,label='$t_0$'),y0 = slider(ymin,ymax,1/10,(ymin+ymax)/2,label='$y_0$'),
soln_curve=checkbox(default=False,label='Draw Solution?')):
G = plot_slope_field(f(t,y),(t,tmin,tmax),(y,ymin,ymax),color='gray')
G += point((t0,y0),color='red',size=40)
G += plot(f(t0,y0)*(t-t0)+y0,(t,t0-0.2,t0+0.2),color='blue') # could blow up if vertical...need parametric
if soln_curve:
desolve_rk4(t^2+y^2,y,ics=[t0,y0],end_points=[tmin,tmax],step=0.1)
# G += points(desolve_rk4(ff(t,yy),yy,ics=[t0,y0],end_points=[tmin,tmax],step=0.1))
G.show(aspect_ratio=1)
html('<center>Slope field for $dy/dt = %s$</center>'%str(f(t,y)))
Click to the left again to hide and once more to show the dynamic interactive window |
# The following still have issues. Fixes?
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# This works ok so long as DE is autonomous. Putting any t's in the DE make this take forever.
P=desolve_rk4(DE,y,ics=[0,1],ivar=t,output='slope_field',end_points=[-4,6],thickness=3)
show(P)
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points(desolve_rk4(f(t,y),y,ics=[0,1]),ivar=t)
verbose 0 (154: primitive.py, options) WARNING: Ignoring option 'ivar'=t
verbose 0 (154: primitive.py, options)
The allowed options for Point set defined by 101 point(s) are:
alpha How transparent the point is.
faceted If True color the edge of the point.
hue The color given as a hue.
legend_color The color of the legend text
legend_label The label for this item in the legend.
rgbcolor The color as an RGB tuple.
size How big the point is (i.e., area in points^2=(1/72
inch)^2).
zorder The layer level in which to draw
verbose 0 (154: primitive.py, options) WARNING: Ignoring option 'ivar'=t
verbose 0 (154: primitive.py, options)
The allowed options for Point set defined by 101 point(s) are:
alpha How transparent the point is.
faceted If True color the edge of the point.
hue The color given as a hue.
legend_color The color of the legend text
legend_label The label for this item in the legend.
rgbcolor The color as an RGB tuple.
size How big the point is (i.e., area in points^2=(1/72
inch)^2).
zorder The layer level in which to draw
verbose 0 (154: primitive.py, options) WARNING: Ignoring option 'ivar'=t
verbose 0 (154: primitive.py, options)
The allowed options for Point set defined by 101 point(s) are:
alpha How transparent the point is.
faceted If True color the edge of the point.
hue The color given as a hue.
legend_color The color of the legend text
legend_label The label for this item in the legend.
rgbcolor The color as an RGB tuple.
size How big the point is (i.e., area in points^2=(1/72
inch)^2).
zorder The layer level in which to draw
verbose 0 (154: primitive.py, options) WARNING: Ignoring option 'ivar'=t
verbose 0 (154: primitive.py, options)
The allowed options for Point set defined by 101 point(s) are:
alpha How transparent the point is.
faceted If True color the edge of the point.
hue The color given as a hue.
legend_color The color of the legend text
legend_label The label for this item in the legend.
rgbcolor The color as an RGB tuple.
size How big the point is (i.e., area in points^2=(1/72 inch)^2).
zorder The layer level in which to draw
verbose 0 (154: primitive.py, options) WARNING: Ignoring option 'ivar'=t
verbose 0 (154: primitive.py, options)
The allowed options for Point set defined by 101 point(s) are:
alpha How transparent the point is.
faceted If True color the edge of the point.
hue The color given as a hue.
legend_color The color of the legend text
legend_label The label for this item in the legend.
rgbcolor The color as an RGB tuple.
size How big the point is (i.e., area in points^2=(1/72 inch)^2).
zorder The layer level in which to draw
verbose 0 (154: primitive.py, options) WARNING: Ignoring option 'ivar'=t
verbose 0 (154: primitive.py, options)
The allowed options for Point set defined by 101 point(s) are:
alpha How transparent the point is.
faceted If True color the edge of the point.
hue The color given as a hue.
legend_color The color of the legend text
legend_label The label for this item in the legend.
rgbcolor The color as an RGB tuple.
size How big the point is (i.e., area in points^2=(1/72 inch)^2).
zorder The layer level in which to draw
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# do something with isoclines...
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# Playing around with slope fields
@interact
def _(t0 = slider(tmin,tmax,1/10,tmin,label='$t_0$'),y0 = slider(ymin,ymax,1/10,(ymin+ymax)/2,label='$y_0$'),
soln_curve=checkbox(default=False,label='Draw Solution?')):
G = plot_slope_field(f(t,y),(t,tmin,tmax),(y,ymin,ymax),color='gray')
G += point((t0,y0),color='red',size=40)
G += plot(f(t0,y0)*(t-t0)+y0,(t,t0-0.2,t0+0.2),color='blue') # could blow up if vertical...need parametric
if soln_curve:
desolve_rk4(t^2+y,y,ics=[t0,y0],end_points=[tmin,tmax],step=0.1)
G += points(desolve_rk4(ff(t,yy),yy,ics=[t0,y0],end_points=[tmin,tmax],step=0.1))
G.show(aspect_ratio=1)
html('<center>Slope field for $dy/dt = %s$</center>'%str(f(t,y)))
Click to the left again to hide and once more to show the dynamic interactive window |
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