# A.5 Well-Posed Problems and Mathematical Models

In addition to uniqueness, another consequence of the inequality in (45) is the fact that solutions of the differential equation

depend *continuously* on the initial value `x`(`a`); that is, if ${x}_{1}(t)$ and ${x}_{2}(t)$ are two solutions of (47) on the interval $a\leqq t\leqq T$ such that the initial values ${x}_{1}(a)$ and ${x}_{2}(a)$ are sufficiently close to one another, then the values of ${x}_{1}(t)$ and ${x}_{2}(t)$ remain close to one another. In particular, if $|{x}_{1}(a)-{x}_{2}(a)|\leqq \delta ,$ then (45) implies that

for all `t` with $a\leqq t\leqq T.$ Obviously, we can make $\u03f5$ as small as we wish by choosing 1 sufficiently close to zero.

This continuity of solutions of (47) with respect to initial values is important in practical ...

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