145_pdfsam_math 54 differential equation solutions odd

145_pdfsam_math 54 differential equation solutions odd -...

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Exercises 3.6 In order to approximate the solution φ ( x )= e x at the point x =1w i th N steps, we take h =( x x 0 ) /N =1 /N ,andso N =1 /h . Then the above formula becomes y N = ± 1+ h/ 2 1 h/ 2 ² N y 0 = ± 1+ h/ 2 1 h/ 2 ² N = ± 1+ h/ 2 1 h/ 2 ² 1 /h and hence e = φ (1) y N = ± 1+ h/ 2 1 h/ 2 ² 1 /h . Substituting h =10 k , k =0 , 1 , 2 , 3, and 4, we Fll in Table 3-A. Table 3–A : Approximations ± 1+ h/ 2 1+ h/ 2 ² 1 /h to e 2 . 718281828 ... . h Approximation Error 1 3 0.281718172 10 1 2.720551414 0.002269586 10 2 2.718304481 0.000022653 10 3 2.718282055 0.000000227 10 4 2.718281831 0.000000003 These approximations are better then those in Tables 3.4 and 3.5 of the text.
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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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