Answer:
To apply the fourth-order Runge-Kutta method, we can use the following steps:
Given:
Initial condition: y = 1 when x = 0
Step size: h = 0.1
We want to find the approximate value of y when x = 1/5.
The fourth-order Runge-Kutta method involves the following steps:
Set up the following equations:
k1 = hf(x, y)
k2 = hf(x + h/2, y + k1/2)
k3 = hf(x + h/2, y + k2/2)
k4 = hf(x + h, y + k3)
Update the values:
y = y + (k1 + 2k2 + 2k3 + k4)/6
x = x + h
Repeat steps 1 and 2 until the desired x-value is reached.
Let's apply these steps to find the approximate value of y when x = 1/5:
Step 1:
x = 0, y = 1
h = 0.1
k1 = 0.1 * f(x, y) = 0.1 * f(0, 1)
k2 = 0.1 * f(x + h/2, y + k1/2) = 0.1 * f(0 + 0.1/2, 1 + k1/2)
k3 = 0.1 * f(x + h/2, y + k2/2) = 0.1 * f(0 + 0.1/2, 1 + k2/2)
k4 = 0.1 * f(x + h, y + k3) = 0.1 * f(0 + 0.1, 1 + k3)
Step 2:
y = y + (k1 + 2k2 + 2k3 + k4)/6
x = x + h
Repeat steps 1 and 2 until x = 1/5.
Performing these calculations step by step will yield the approximate value of y when x = 1/5 using the fourth-order Runge-Kutta method.
Explanation: