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Example 3 - Explosion

y = y2, x(0) = 1

may be solved by separation of variables:

14_differential_equations-111.gif

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.

14_differential_equations-112.gif

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.

Δt = 0.2

t

y(t)

Y(t)

Y(t)2 Δt

0.0

1.0

1.0

0.2

0.2

1.25

1.2

0.288

0.4

1.6667

1.4488

0.4428

0.6

2.5

1.9309

0.7456

0.8

5.0

2.6764

1.4327

1.0

4.1091

3.3770

1.2

7.4861

11.2084

1.4

18.6945

69.8966

1.6

88.5910

1569.6736

1.8

1658.26

549968.3

2.0

551627.6


Last Update: 2010-11-25