TAILIEUCHUNG - A textbook of Computer Based Numerical and Statiscal Techniques part 37

A textbook of Computer Based Numerical and Statiscal Techniques part 37. 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. | 346 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES - -------------- y4 at x4 x0 4h 0 Hence y . Ans. Example 18. Given d- x y with initial condition y 0 1. Find y .05 and y .1 correct to 6 decimal places. Sol. Using Euler s method we obtain y1 0 y y hf fo y 1 .05 0 1 We improve y1 by using Euler s modified method y 1 y0 2 f x0 y0 f x1 y 1 .05 0 1 .05 . y 2 1 505 0 1 .05 y13 1 505 0 1 .05 y14 1 J05 0 1 .05 . Since y 3 y 4 correct to 6 decimal places. Hence we take y1 . we have y .05 Again using Euler s method we obtain y20 y2 y hf x1 y .05 .05 . We improve y2 by using Euler s modified method y21 y. .05 .1 . y22 505 .05 .1 . NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATION 347 y23 .05 .1 . y24 y. .05 .1 . y25 .05 .1 . Since y24 y25 correct to 7 decimal places. Hence we take y2 . Therefore we have y .1 correct to 6 decimal places. Ans. Example 19. Find y using Euler s method for -xy2 where y 2 1. Take h .1 dx Sol. By Euler s method we obtain y 0 y1 y0 hf x0 y0 1 .1 -2 -1 2 .8. This value of y1 is improved by using Euler s modified method y 1 y0 h f x0 y0 f x1 y 0 1 01 -2 1 2 .8 2 .8328 Similarly y12 1 -2 1 2 .8328 2 .8272 y 3 1 01 -2 1 2 .8272 2 .8281 y 4 1 01 -2 1 2 .8281 2 .8280 y 5 1 01 -2 1 2 .8280 2 .8280 Since y 4 y 5 . Hence we take y1 .828 at x1 Now if y2 is the value of y at x . Then we apply Euler s method to compute y . we obtain y20 y2 y hf x1 y1 .828 .1 .828 2 .68402 348 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Now using Euler s modified formula we obtain y 21 .828 01

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