βh 2 α n 1 α 1 From the definition of α 1 hL it is obvious that E n βh 1 hL n 1

# Βh 2 α n 1 α 1 from the definition of α 1 hl it

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βh 2 α n - 1 α - 1 . From the definition of α = 1 + hL , it is obvious that | E n | ≤ βh (1 + hL ) n - 1 L .

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This error is clearly increasing exponentially in n . c. Consider the function g ( x ) = ln(1 + x ) for x > 0, then g 0 ( x ) = 1 1+ x < 1. It follows that ln(1 + x ) x for all x 0. Thus, ln(1 + hL ) hL, ln(1 + hL ) n nhL, (1 + hL ) n e nhL . Thus, | E n | ≤ βh e nhL - 1 L . For T > t 0 and taking h such that n steps are required to reach T , then nh = T - t 0 , so | E n | ≤ βh e ( T - t 0 ) L - 1 L = Kh, where K depends on the length of the interval and the constants L and β , which came from properties of f . 8. (9pts) c. The differential equation, y 0 = y 2 / 3, has a vertical asymptote for finite t ( t = 3) depending on the initial condition, y (0) = 1. (Different versions have different asymptotes.) The solutions track well for the early part of the interval, but lose accuracy as t approaches the asymptote. The smaller stepsizes improve the computations. However, Improved Euler’s method does much better at tracking the actual solution for a longer time. 0 0.5 1 1.5 2 2.5 3 t 0 5 10 15 20 25 30 y Actual Solution Euler h =0 . 1 Euler h =0 . 05 0 0.5 1 1.5 2 2.5 3 t 0 20 40 60 80 100 120 140 y Actual Solution Im Euler h =0 . 1 Im Euler h =0 . 05 Euler simulation Improved Euler simulation f. The graph of this differential equation ( P 0 = (1 . 46 - 0 . 54 t ) P ) again shows that the Improved Euler’s method is significantly better at tracking the actual solution. Thus, the maximum is
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