27_pdfsam_math 54 differential equation solutions odd

27_pdfsam_math 54 differential equation solutions odd -...

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Exercises 1.4 For approximation of φ ( t )a tth epo in t t =1w i th N = 20 steps, we take the step size h =(1 t 0 ) / 20 = 0 . 05. Thus, the recursive formulas for Euler’s method are t n +1 = t n +0 . 05 , x n +1 = x n +0 . 05 ( 1+ x 2 n ) . Applying these formulas with n =0 , 1 ,..., 19, we obtain x 1 = x 0 +0 . 05 ( 1+ x 2 0 ) =0 . 05 , x 2 = x 1 +0 . 05 ( 1+ x 2 1 ) =0 . 05 + 0 . 05 ( 1+0 . 05 2 ) =0 . 100125 , x 3 = x 2 +0 . 05 ( 1+ x 2 2 ) =0 . 100125 + 0 . 05 ( 1+0 . 100125 2 ) 0 . 150626 , . . . x 19 = x 18 +0 . 05 ( 1+ x 2 18 ) 1 . 328148 , φ (1) x 20 = x 19 +0 . 05 ( 1+ x 2 19 ) =1 . 328148 + 0 . 05 ( 1+1 . 328148 2 ) 1 . 466347 , which is a good enough approximation to φ (1) = tan 1 1 . 557408. 13. From Problem 12, y n =(1+1 /n ) n and so lim n →∞ [( e y n ) / (1 /n )] is a 0 / 0 indeterminant. If we let h =1 /n in y n and use L’Hospital’s rule, we get lim n →∞ e
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This note was uploaded on 03/29/2010 for the course MATH 54257 taught by Professor Hald,oh during the Spring '10 term at Berkeley.

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