2.6 CONTINUATION OF SOLUTIONS

The exact continuous solution of Example 2 is shown in Figure 2.3 for the interval ? ? < x < 1, where the domain (i.e., the region where a unique continuous solution can be proven to exist by the existence theorem) is also indicated. Since is substantially smaller than the full interval, it appears that the conditions of the existence theorem restrict the actual domain of validity of the solution. This is true in many cases, but it should be apparent that we can continue the solution beyond both sides of by taking new ICs

and

for the ODE

For example, if we do this for , the largest domain around is found to be

so that existence is now proven for . Similarly, the solution can also be continued to the left of .

A natural question now arises: How far can the solution be continued? The answer is contained in the following result (cf. Brauer and Nohel, 1989, section 3.4; Coddington, 1989, chapter 5, section 7).

Theorem

Let f( x, y) be continuous and bounded, and let is satisfy a Lipschitz condition in region . Then for every ( x ₀, y ₀) ? , the ODE

possesses a unique solution y( x) satisfying y( x ₀) = y ₀, which can be extended in both directions of x ₀ until it remains finite or until its...