chapter5.page08

chapter5.page08 - 2 1 2 k 2 = 0 25 f(0 625 750224 0 5...

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8 1. NUMERICAL METHODS FOR ORDINARY DIFFERENTIAL EQUATIONS x 1 : k 1 = hf ( t 0 ,x 0 ) = 0 . 25 * f (0 , 1) = 0 k 2 = hf ( t 0 + 1 2 h,x 0 + 1 2 k 1 ) = f (0 . 125 , 1) = - 0 . 373276 k 3 = hf ( t 0 + 1 2 h,x 0 + 1 2 k 2 ) = 0 . 25 f (0 . 125 , 1 + 0 . 5( - 0 . 373276)) = - 0 . 373276 k 4 = hf ( t 0 + h,x 0 + k 3 ) = 0 . 25 f (0 . 25 , 1 - 0 . 373276) = - 0 . 309854 x 1 = x 0 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) = 1 + 1 6 (0 + 2 * ( - 0 . 373276) + 2 * ( - 0 . 373276) - 0 . 309854) = 0 . 923911 t 1 = t 0 + h = 0 + 0 . 25 = 0 . 25 x 2 : k 1 = hf ( t 1 ,x 1 ) = 0 . 25 * f ( . 25 , 0 . 923911) = - 0 . 560741 k 2 = hf ( t 1 + 1 2 h,x 1 + 1 2 k 1 ) = f (0 . 125 , 0 . 923911 + 0 . 5( - 0 . 560741)) = - 0 . 719601 k 3 = hf ( t 1 + 1 2 h,x 1 + 1 2 k 2 ) = 0 . 25 f (0 . 125 , 0 . 923911 + 0 . 5( - 0 . 719601)) = - 0 . 702688 k 4 = hf ( t 1 + h,x 1 + k 3 ) = 0 . 25 f (0 . 5 , 0 . 923911 - 0 . 702688) = - 0 . 763158 x 2 = x 1 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) = 0 . 923911 + 1 6 ( - 0 . 560741 + 2 * ( - 0 . 719601) + 2 * ( - 0 . 702688) - 0 . 763158) = 0 . 750224 t 2 = t 1 + h = 0 . 25 + 0 . 25 = 0 . 5 x 3 : k 1 = hf ( t 2 ,x 2 ) = 0 . 25 * f ( . 5 , 0 . 750224) = - 0 . 765171 k 2 = hf ( t 2 + 1 2 h,x 2 + 1 2 k 1 ) = f (0 . 625 , 0 . 750224 + 0 . 5( - 0 . 765171)) = - 0 . 735295 k 3 = hf ( t 2 + 1 2 h,x
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Unformatted text preview: 2 + 1 2 k 2 ) = 0 . 25 f (0 . 625 , . 750224 + 0 . 5(-. 735295)) =-. 739508 k 4 = hf ( t 2 + h,x 2 + k 3 ) = 0 . 25 f (0 . 75 , . 750224-. 739508) =-. 637224 x 3 = x 2 + 1 6 ( k 1 + 2 k 2 + 2 k 3 + k 4 ) = 0 . 750224 + 1 6 (-. 765171 + 2 * (-. 735295) + 2 * (-. 739508)-. 637224) = 0 . 568891 t 3 = t 2 + h = 0 . 5 + 0 . 25 = 0 . 75...
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