8.4 8.5

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

def system(t, state):
    x, y = state
    dxdt = -x*3 - y
    dydt = x - y*3
    return [dxdt, dydt]

t_span = (0, 30)
y0 = [1, 0.5]

sol = solve_ivp(system, t_span, y0, t_eval=np.linspace(t_span[0], t_span[1], 1000))

x = sol.y[0]
y = sol.y[1]
t = sol.t

plt.figure(figsize=(12, 6))

plt.subplot(1, 2, 1)
plt.plot(t, x, label='x(t)')
plt.xlabel('t')
plt.ylabel('x(t)')
plt.title('x(t) vs t')
plt.legend()
plt.grid(True)

plt.subplot(1, 2, 2)
plt.plot(t, y, label='y(t)')
plt.xlabel('t')
plt.ylabel('y(t)')
plt.title('y(t) vs t')
plt.legend()
plt.grid(True)

plt.figure(figsize=(6, 6))
plt.plot(x, y, label='Phase Trajectory')
plt.xlabel('x')
plt.ylabel('y')
plt.title('Phase Plane (x, y)')
plt.legend()
plt.grid(True)
plt.axis('equal')

plt.tight_layout()
plt.show()

print("3010")

import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp

def model(t, y):
    f, df_dm, d2f_dm2, T, dT_dm = y
    d3f_dm3 = -3*f*d2f_dm2 + 2*(df_dm)**2 - T
    d2T_dm2 = -2.1*f*dT_dm
    return [df_dm, d2f_dm2, d3f_dm3, dT_dm, d2T_dm2]

y0 = [0, 0, 0.68, 1, -0.5]

t_span = (0, 10)
t_eval = np.linspace(t_span[0], t_span[1], 1000)

sol = solve_ivp(model, t_span, y0, t_eval=t_eval, method='RK45')

f = sol.y[0]
T = sol.y[3]

plt.figure(figsize=(12, 6))

plt.subplot(2, 1, 1)
plt.plot(sol.t, f, label='f(m)')
plt.xlabel('m')
plt.ylabel('f(m)')
plt.title('Solution for f(m)')
plt.grid(True)

plt.subplot(2, 1, 2)
plt.plot(sol.t, T, label='T(m)', color='orange')
plt.xlabel('m')
plt.ylabel('T(m)')
plt.title('Solution for T(m)')
plt.grid(True)

plt.tight_layout()
plt.show()

print("3010")