How can I edit my 4th Order Runge Kutta code to evaluate a 2nd order ODE?

Question

Python question:

I have the following RK-4 code that evaluates my first order ode, but I want to edit it such that it can evaluate the following 2nd order ode as a system of first order ODE's: (d^2y/dt^2) +4(dy/dt)+2y=0. I would like to keep h (the step size the same) and the initial conditions would be y(0)=1 and y'(0)=3.

I am very new to python, so I am having a hard time doing this. Thank you in advance for any help!

import matplotlib.pyplot as plt 

def dydx(x, y): 
    return ((x - y)/2)

def rungeKutta(x0, y0, x, h): 
   
  n = (int)((x - x0)/h)  
  
  y = y0 
  for i in range(1, n + 1): 
 
    k1 = h * dydx(x0, y) 
    k2 = h * dydx(x0 + 0.5 * h, y + 0.5 * k1) 
    k3 = h * dydx(x0 + 0.5 * h, y + 0.5 * k2) 
    k4 = h * dydx(x0 + h, y + k3) 
    
    y = y + (1.0 / 6.0)*(k1 + 2 * k2 + 2 * k3 + k4)   
    x0 = x0 + h 

  return y 
  
x0 = 0
y = 1
x = 2
h = 0.2

print ('The value of y at x is:', rungeKutta(x0, y, x, h))

Answer 1

The only change that is necessary is to make the state a vector object. In python this is usually a numpy array. So

y0 = np.array([1,3],dtype=float)
def dydx(x, y): 
    return np.array([y[1], -4*y[1]-2*y[0]])

Everything else should automatically use the appropriate vector operations.