Runge-Kutta Third Order Method

The Runge-Kutta method is a numerical technique used to solve ordinary differential equations. It is an iterative method that approximates the solution by calculating intermediate values at various points. The third-order Runge-Kutta method is a specific implementation of this technique.

Given Differential Equation

The given differential equation is dy/dx = -x.

Initial Condition

We are given that y(2) = 1. This means that the value of y at x = 2 is 1.

Step Size

The step size is the difference between the x-values of two consecutive points. In this case, the step size is 0.2 because we need to find the value of y at x = 2.2.

Runge-Kutta Third Order Method Algorithm

The third-order Runge-Kutta method involves the following steps:

1. Initialize the initial condition: y(2) = 1.
2. Calculate the intermediate values using the following formulas:
a. k1 = h * f(x, y)
b. k2 = h * f(x + h/2, y + k1/2)
c. k3 = h * f(x + h, y - k1 + 2k2)
3. Calculate the new value of y:
y(2.2) = y(2) + (k1 + 4k2 + k3)/6, where h is the step size.

Calculation

Using the given differential equation dy/dx = -x, we can determine f(x, y) as -x.

Following the algorithm of the third-order Runge-Kutta method:

1. k1 = 0.2 * (-2) = -0.4
2. k2 = 0.2 * (-(2 + 0.2/2)) = -0.42
3. k3 = 0.2 * (-(2 + 0.2) - (-0.4) + 2(-0.42)) = -0.48

Now, we can calculate the new value of y:

y(2.2) = 1 + (-0.4 + 4(-0.42) - 0.48)/6
= 1 - 0.21
= 0.79

Therefore, the value of y(2.2) using the Runge-Kutta third-order method is approximately 0.79.

Conclusion

By applying the third-order Runge-Kutta method to the given differential equation with the initial condition, we obtained the value of y at x = 2.2 as 0.79. This matches with option 'C' as the correct answer.

Here, f(x) = −x + 1