Solution eulers method uses w y 0 w i 1 w i hf t i w

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Solution: Euler’s method uses: ( w 0 = y (0) w i +1 = w i + hf ( t i , w i ) , which in the case of our problem gives w 0 = e and w i +1 = w i + h · - 9 w i = (1 - 9 h ) w i = (1 - 9 · 0 . 1) w i = 0 . 1 w i . Of course we can compute these w i iteratively, but also it is not hard to see that the general formula for w i +1 is e 10 i and this may be a bit easier to do by hand. The table below expresses these approximations compared to the actual solution. i t i w i y ( t i ) 0 0 2.71828 2.71828 1 0.1 0.271828 1.10517 2 0.2 0.0271828 0.449329 3 0.3 0.00271828 0.182684 4 0.4 0.000271828 0.0742736 5 0.5 0.0000271828 0.0301974 6 0.6 2 . 71828 × 10 - 6 0.0122773 7 0.7 2 . 71828 × 10 - 7 0.00499159 8 0.8 2 . 71828 × 10 - 8 0.00202943 9 0.9 2 . 71828 × 10 - 9 0.000825105 10 1 2 . 71828 × 10 - 10 0.000335463
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