Differential Equations and Linear Algebra, 4th Edition

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

$$\frac{dx}{dt}=f(x,t)$$ (47)

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

$$|x_1(t)-x_2(t)|\leqq \delta e^{k(T-a)}=ϵ$$ (48)

for all t with $a\leqq t\leqq T.$ Obviously, we can make $ϵ$ 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 ...