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 # Q 3A


# d2x/dt2 + gamma * dx/dt + kx = Fcos wt

# Exact Sol of Amplitude = F / sqrt((k-w)^2 + gamma^2*w^2)

# take gamma = 0.5 , k = 5 , F = 1



import numpy as np

import matplotlib.pyplot as plt

from scipy.integrate import odeint


k  = 5

gam = 0.5

F = 1.


def forced(x ,t):

    return [x[1], F*np.cos(w*t) - gam*x[1] - k*x[0]]



x0 = [0., 1.]



t = np.linspace(0, 100, 1001)

freq = np.linspace(0.5, 5, 1001)



A = []

for w in freq:

    sol = odeint(forced, x0, t)

    X = sol[:,0]

    amp = np.max(X[-200:])

    A.append(amp)


exact = F/(np.sqrt((k-freq**2)**2 + gam**2*freq**2))


    

plt.plot(freq[::2], A[::2], '--', label="soultion by odeint")

plt.plot(freq, exact, 'o', ms=0.5, label ="exact solution")

plt.xlabel("frequency")

plt.ylabel("Amplitude")

plt.legend()

plt.show()

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