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149_pdfsam_math 54 differential equation solutions odd

149_pdfsam_math 54 differential equation solutions odd -...

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Exercises 3.6 0 0.5 1 1.5 2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 y x Figure 3–B : Polygonal line approximation to the solution of y = x + 3 cos( xy ), y (0) = 0. A polygonal line, approximating the graph of the solution to the given initial value problem, which has vertices at points ( x, y ) from Table 3-C, is sketched in Figure 3-B. 13. We want to approximate the solution φ ( x ) to y = 1 y + y 3 , y (0) = 0, at x = 1. (In other words, we want to find an approximate value for φ (1).) To do this, we will use the algorithm on page 130 of the text. (We assume that the reader has a programmable calculator or microcomputer available and can transform the step-by-step outline on page 130 into an
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Unformatted text preview: executable program. Alternatively, the reader can use the software provided free with the text.) The inputs to the program are x = 0, y = 0, c = 1, ε = 0 . 003, and, say, M = 100. Notice that by Step 6 of the improved Euler’s method with tolerance, the computations should terminate when two successive approximations di²er by less that 0 . 003. The initial value for h in Step 1 of the improved Euler’s method subroutine is h = (1 − 0)2 − = 1 . For the given equation, we have f ( x, y ) = 1 − y + y 3 , and so the numbers F and G in Step 3 145...
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