import numpy as np
import scipy.integrate as integr
import matplotlib.pyplot as plt
# import scipy.optimize as resol

def trace(a,b,u,q) :
    def f(X,t) :
        y,z = X
        return [z,-(1+q(t))*y]
    T = np.arange(0,u,0.01)
    Y = integr.odeint(f,[a,b],T)
    plt.plot(T,Y[:,0])
    plt.show()
    
def q1(t) :
    return -t**2/(1+t**2)
    
def q2(t) :
    return 1/((1+t)**0.5)

def q3(t) :
    return 1/(1+t**2)