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Apply fourth-order Runge-Kutta method to find an approximate value of y when x = 1/5, given that


(dy)/(dx) = x + y
y = 1, when x = 0 with h = 0.1​

User Mdemolin
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1 Answer

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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:

User Acushner
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