295_03-04_Complex_Eigenvales_Notes_Example_01.sagews

# declare the variables
x, y, t = var('x y t')


# declare the system
F = [2*x - 6*y, 2*x + y]


# create the vector field
# normalize the vector fields so that all of the arrows are the same length
n = sqrt(F[0]^2 + F[1]^2)
# plot the vector field
p030401 = plot_vector_field((F[0]/n, F[1]/n), (x, -4, 4), (y, -4, 4), aspect_ratio = 1)


# plot the straightline solutions
# p03 += plot(-1/3*x, x, -4, 4, thickness = 2, color = 'red', xmin = -4, xmax = 4, ymin = -4, ymax = 4)
# p03 += plot(x, x, -4, 4, thickness = 2, color = 'red')


# solve the system for the initial condition t = 0, x = 1, y = 1
P1 = desolve_system_rk4(F, [x, y], ics=[0, 1, 1], ivar = t, end_points = [-5,5], step = 0.01)

# grab the x and y values from the set P1
S1 = [ [j, k] for i, j, k in P1]

# plot the solution
p030401 += line(S1, thickness = 2, axes_labels=['$x(t)$','$y(t)$'], xmin = -4, xmax = 4, ymin = -4, ymax = 4, color = 'blue')


# solve the system for the initial condition t = 0, x = -1, y = 1
P2 = desolve_system_rk4(F, [x, y], ics=[0, -1, 1], ivar = t, end_points = [-5,5], step = 0.01)

# grab the x and y values from the set P2
S2 = [ [j, k] for i, j, k in P2]

# plot the solution
p030401 += line(S2, thickness = 2, axes_labels=['$x(t)$','$y(t)$'], xmin = -4, xmax = 4, ymin = -4, ymax = 4, color = 'purple')


# solve the system for the initial condition t = 0, x = 3, y = 0
P3 = desolve_system_rk4(F, [x, y], ics=[0, 3, 0], ivar = t, end_points = [-5,5], step = 0.01)

# grab the x and y values from the set P3
S3 = [ [j, k] for i, j, k in P3]

# plot the solution
p030401 += line(S3, thickness = 2, axes_labels=['$x(t)$','$y(t)$'], xmin = -4, xmax = 4, ymin = -4, ymax = 4, color = 'green')


# display the final plot
show(p030401)

# Save the plot as a pdf to use in LaTeX or Word or your program of choice.
p030401.save('295_03-04_Notes_Example_01.pdf')