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163_pdfsam_math 54 differential equation solutions odd

163_pdfsam_math 54 differential equation solutions odd - x...

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Exercises 3.7 On the 82nd step we get x = 1 . 405 + 0 . 005 = 1 . 410 , y = 0 . 004425 + 1 6 (0 . 002566 + 2 · 0 . 002548 + 2 · 0 . 002548 + 0 . 002530) ≈ − 0 . 001876 , and the next step gives k 1 = 0 . 005[2(1 . 410) 4 ( 0 . 001876) 2 ] = 0 . 002530 ; k 2 = 0 . 005[2(1 . 410 + 0 . 005 / 2) 4 ( 0 . 001876 + 0 . 002530 / 2) 2 ] = 0 . 002512 ; k 3 = 0 . 005[2(1 . 410 + 0 . 005 / 2) 4 ( 0 . 001876 + 0 . 002512 / 2) 2 ] = 0 . 002512 ; k 4 = 0 . 005[2(1 . 410 + 0 . 005) 4 ( 0 . 001876 + 0 . 002512) 2 ] = 0 . 002494 ; x = 1 . 410 + 0 . 005 = 1 . 415 , y = 0 . 414 + 1 6 (0 . 002530 + 2 · 0 . 002512 + 2 · 0 . 002512 + 0 . 002494) 0 . 000636 . Since y (1 . 41) < 0 and y (1 . 415) > 0 we conclude that the root of the solution is on the interval (1 . 41 , 1 . 415). As a check, we apply the 4th order Runge-Kutta subroutine to approximate the solution to the given initial value problem on [1 , 1 . 5] with a step size h = 0 . 001, which requires N = (1 . 5 1) / 0 . 001 = 500 steps. This yields y (1 . 413) ≈ − 0 . 000367, y (1 . 414) 0 . 000134, and so, within two decimal places of accuracy,
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Unformatted text preview: x ≈ 1 . 41 . 13. For this problem f ( x, y ) = y 2 − 2 e x y + e 2 x + e x . We want to ±nd the vertical asymptote located in the interval [0 , 2] within two decimal places of accuracy using the Forth Order Runge-Kutta subroutine. One approach is to use a step size of 0 . 005 and look for y to approach in±nity. This would require 400 steps. We will stop the subroutine when the value of y (“blows up”) becomes very large. For Step 3 we have k 1 = hf ( x, y ) = 0 . 005 ( y 2 − 2 e x y + e 2 x + e x ) , k 2 = hf ± x + h 2 , y + k 1 2 ² = 0 . 005 ± y + k 1 2 ² 2 − 2 e ( x + h/ 2) ± y + k 1 2 ² + e 2( x + h/ 2) + e ( x + h/ 2) , 159...
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