ODEGraph

import numpy as np
from scipy.integrate import odeint
from scipy.integrate import quad
def f(y,x):
    dydx = -x/y
    return dydx

g = lambda x: -(-x**2 + 25)**(0.5)
def RK4OdeSys(f,c,x):
    H = x[1] - x[0]

    N = len(x)

    z = np.zeros(N)
    z[0] = c


    for k in range(0,N-1):
        k1 = f(z[k],x[k])

        k2 = f(z[k]+ ((H) * (k1/2)), x[k] + (H/2) )

        k3 = f(z[k]+ (H * (k2/2)), x[k] + (H/2) )

        k4 = f(z[k]+ (H * k3) , x[k] + H )

        z[k+1] =  z[k] + ((H/6) * (k1 + 2*k2 + 2*k3 + k4) )
    return z

H = 0.01
x =  np.arange(4,5,H)

z0 = -3

z_odeint =  odeint(f,z0,x)

z_rk_4 =  RK4OdeSys(f,z0,x)

z_actual = g(x)
z_odeint = z_odeint.reshape(100,)

fig = plt.figure(1,figsize=(8,5))
er1 = abs(z_actual - z_odeint)
plt.plot(x,er1)
er2 = abs(z_actual - z_rk_4)
plt.plot(x,er2,"r")
plt.xlabel("x")
plt.ylabel("y")
plt.legend(["exact vs odeint","Exact vs Rung-Kutta"])
plt.title("Comparision of approximation errors")