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Differential Equations Solutions 111

Differential Equations Solutions 111 - 121 The last two...

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121 The last two terms can be combined and represent an effective local error. Therefore, the global error is magnified if | (1 ha/ 2) / (1 + ha/ 2) | > 1. Conversely, the method is stable when 1 ha/ 2 1 + ha/ 2 < 1 , which holds for all h > 0. CHALLENGE 20.7. Recall that Euler’s method is y n +1 = y n + hf ( t n , y n ) , and backward Euler is y n +1 = y n + hf ( t n +1 , y n +1 ) . P : y = 1 + . 1(1 2 ) = 1 . 1 , E : f = (1 . 1) 2 . 5 = 0 . 71 , C : y = 1 + . 1 . 71 = 1 . 071 , E : f = (1 . 071) 2 0 . 5 . The predicted value is quite close to the corrected value; this is an indication that the stepsize is small enough to obtain some accuracy in the computed solution. CHALLENGE 20.8. f ( t, y ) = 10 y 2 20. P: y P = y (0) + . 1 f (0 , y (0)) = 1 + . 1( 10) = 0 . E: f P = f ( . 1 , y P ) = 10
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