Question

Use python (spi.RK45 )to code the Runge Kutta method to approximate/plot the solution the following initial-value

?′=1+(?−?)2, 2<?<3, ?(2)=1y′=1+(t−y)2, 2

Answer 1

def dydx(t, y): 
    return 1+(t-y)*2 

def rungeKutta(x0, y0, x, h): 
    # Count number of iterations using step size or 
    # step height h 
    n = (int)((x - x0)/h)  
    # Iterate for number of iterations 
    y = y0 
    for i in range(1, n + 1): 
        "Apply Runge Kutta Formulas to find next value of y"
        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) 
  
        # Update next value of y 
        y = y + (1.0 / 6.0)*(k1 + 2 * k2 + 2 * k3 + k4) 
  
        # Update next value of x 
        x0 = x0 + h 
    return y 
  
# Driver method 
x0 = 0
y = 1
t = 2
h = 0.2
print ('The value of y at x is:', rungeKutta(x0, y, x, h))