Plotting Direction Fields for Differential Equations
Description
We will plot the direction field for $dy/dt = 2y(1- y)$ for $-2 < t < 2$ and $-2 < y < 2$.
Sage Cell
Code
y, t = var('y, t')
f = 2*y*(1 - y)
x_min = -2 #set the window size
x_max = 2
y_min = -2
y_max = 2
#plot the slope field and display
p = plot_slope_field(f, (t, x_min, x_max), (y, y_min, y_max), headaxislength=3, headlength=3, axes_labels = ['$t$','$y$'], xmin = x_min, xmax = x_max, ymin = y_min, ymax = y_max)
pOptions
Option
We add three solution curves to the plot ($y(0) = 0.5$, $y(0) = 1.01$, and $y(0) = -0.01$).
Code
y, t = var('y, t')
f = 2*y*(1 - y)
x_min = -2 #set the window size
x_max = 2
y_min = -2
y_max = 2
t_min = -2 #set the min and max for t
t_max = 4
t_start = 0 #set the initial conditions
y_start = 0.5
#plot first solution
P = desolve_rk4(f, y, ics=[t_start, y_start], ivar = t, end_points=[t_min, t_max], thickness=3)
p = line(P, title="$y' = 2y(1-y)$")
#plot second solution
y_start = 1.01
P = desolve_rk4(f, y, ics=[t_start, y_start], ivar = t, end_points=[t_min, t_max], thickness=3)
p += line(P)
#plot third solution
y_start = -0.01
P = desolve_rk4(f, y, ics=[t_start, y_start], ivar = t, end_points=[t_min, t_max], thickness=3)
p += line(P)
#plot the slope field and display
p += plot_slope_field(f, (t, x_min, x_max), (y, y_min, y_max), headaxislength=3, headlength=3, axes_labels = ['$t$','$y$'], xmin = x_min, xmax = x_max, ymin = y_min, ymax = y_max)
pOption
We now add the two equilibrium solutions ($y = 0$ and $y = 2$) to the plot.
Code
y, t = var('y, t')
f = 2*y*(1 - y)
x_min = -2 #set the window size
x_max = 2
y_min = -2
y_max = 2
t_min = -2 #set the min and max for t
t_max = 4
t_start = 0 #set the initial conditions
y_start = 0.5
#plot first solution
P = desolve_rk4(f, y, ics=[t_start, y_start], ivar = t, end_points=[t_min, t_max], thickness=3)
p = line(P, title="$y' = 2y(1-y)$")
#plot second solution
y_start = 1.01
P = desolve_rk4(f, y, ics=[t_start, y_start], ivar = t, end_points=[t_min, t_max], thickness=3)
p += line(P)
#plot third solution
y_start = -0.01
P = desolve_rk4(f, y, ics=[t_start, y_start], ivar = t, end_points=[t_min, t_max], thickness=3)
p += line(P)
#plot equilibrium solutions
p += plot(0, (y, x_min, x_max), color = "red", thickness=2, linestyle="--")
p += plot(1, (y, x_min, x_max), color = "red", thickness=2, linestyle="--")
#plot the slope field and display
p += plot_slope_field(f, (t, x_min, x_max), (y, y_min, y_max), headaxislength=3, headlength=3, axes_labels = ['$t$','$y$'], xmin = x_min, xmax = x_max, ymin = y_min, ymax = y_max)
pTags
Primary Tags: Differential Equations: First order differential equations
Secondary Tags: First order differential equations: Direction fields
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Author: T. Judson
Date: 07 Feb 2019 18:36
Submitted by: Tom Judson