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


# declare the system
F = [-3*x - 2*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
pQuiz09 = plot_vector_field((F[0]/n, F[1]/n), (x, -4, 4), (y, -4, 4), aspect_ratio = 1)


# plot the straightline solutions [-1,1]^T => y = -x
pQuiz09 += plot(-1*x, x, -4, 4, thickness = 2, color = 'red', xmin = -4, xmax = 4, ymin = -4, ymax = 4)


# 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
pQuiz09 += line(S1, thickness = 2, axes_labels=['$x(t)$','$y(t)$'], xmin = -4, xmax = 4, ymin = -4, ymax = 4, color = 'blue')

# display the final plot
show(pQuiz09)

# Save the plot as a pdf to use in LaTeX or Word or your program of choice.
pQuiz09.save('295_03-05_Quiz_09.pdf')