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Example 3 - Explosion
y = y2, x(0) = 1 may be solved by separation of variables: The graph, shown in Figure 14.4.5, approaches infinity as t approaches 1 from the left. The function y(t) = (1 - t)-1 is a solution for 0 ≤ t < 1, and the solution has an explosion at t = 1. Figure 14.4.5 Example 3 The Euler approximation Y(t) with a real increment Δt can be computed even for t greater than 1, but will approach y(t) only for 0 ≤ t < 1. For infinitesimal Δt, the Euler approximation will be finite and infinitely close to y(t) when t is in the real interval [0,1). Y(t) will keep on increasing and will be infinite for all t with standard part ≥ 1. Continuing the example, compute the Euler approximation Y(t) for Δt = 0.2 and 0 ≤ t ≤ 2, and compare the values with the solution y(t) = (1 - t)-1 for 0 ≤ t < 1. The results are shown in the next table and are graphed in Figure 14.4.5.
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Home Differential Equations Existence and Approximation of Solutions Examples Example 3 - Explosion |