TAILIEUCHUNG - A textbook of Computer Based Numerical and Statiscal Techniques part 38

A textbook of Computer Based Numerical and Statiscal Techniques part 38. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 356 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES k4 hf x0 h y0 k3 0 1 - 1 .3 y 2 y0 k1 2k2 2k3 k4 6 1 1 - 0 2 2 - The exact value of y is . Example 3. Solve the equation y x y with y0 1 by Runge-Kutta rule from x 0 to x with h . Sol. Here f x y x y h given y0 1 when x0 0. We have k1 hf x0 y0 0 1 f h k1 k2 hf I x0 y0 2 I f h k2 k3 hf I x0 y0 2 I k4 hf x0 h y0 k3 1 y1 y x y0 k1 2k2 2k3 k4 1 1 111034 6. Similarly for finding y2 y x we get k1 hf x1y1 f h k1 k2 hf I x1 2 y y1 f h k2 k3 hf I x1 2 y y- I NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATION 357 k4 - hf x1 h yi k3 - 1 y2 - y - y1 k1 2k2 2k3 k4 1 2 Similarly for finding y3 y x we get k1 hf x2 y2 I h 1 k2 - hf I X2 y2 - 132428 - I h k2 k3 - hf I X2 2 y2 I - - k4 - hf x2 h y2 k3 - - .-. y3 y x y2 k1 2K2 2K3 k4 Similarly for finding y4 y x we get k1 k2 k3 k4 k1 - k2 - k3 - k1 - 1 y4 - 2 y4 . Ans. Example 4. Given y - x with y 0 2 find y and y correct to 4 decimal places. dx Sol. We have x0 0 y0 2 h Then we get k1 hf x0 y0 2 - 0 358 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES I h 1 k2 hf I X0 y0 2 2 02 0 02 2 Therefore n 2 ----- 2 k4 hf X0 h y0 k3 2 - 0 2 I h 2 k3 hf I X0 y0 y V0 1 k1 2k2 2k3 k4 6 2 y

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