What is the correct formula for Runge-Kutta 3rd order method?
Show Hint
- \( y = y_0 + \frac{1}{6}(k_1 + 2k_2 + 3k_3) \)
- \( y = y_0 + \frac{1}{3}(k_1 + 4k_2 + k_3) \)
- \( y = y_0 + \frac{1}{6}(k_1 + 4k_2 + k_3) \)
- \( y = y_0 + \frac{1}{6}(k_1 + k_2) \)
The Correct Option is A
Solution and Explanation
The Runge-Kutta method is used to approximate solutions of ordinary differential equations (ODEs). The 3rd order Runge-Kutta method (RK3) can be expressed by the formula:
\[ y = y_0 + \frac{1}{6}(k_1 + 2k_2 + 3k_3) \]
Where:
- \( k_1 = h f(x_0, y_0) \)
- \( k_2 = h f(x_0 + \frac{h}{2}, y_0 + \frac{k_1}{2}) \)
- \( k_3 = h f(x_0 + h, y_0 - k_1 + 2k_2) \)
Thus, the correct formula for the Runge-Kutta 3rd order method is:
\[ y = y_0 + \frac{1}{6}(k_1 + 2k_2 + 3k_3) \]
This corresponds to option (A).
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