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 $ϵ$ 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 applications where we are unlikely to know the initial value $x_0=x(a)$ with absolute precision. For example, suppose that the initial value problem

models a population for which we know only that the initial population is within $\delta >0$ of the assumed value $x_0.$ Then even if the function f(x, t) is accurate, the solution x(t) of (49) will be only an approximation to the actual population. But (45) implies that the actual population at time t will be within $\delta e^{k(T-a)}$ of the approximate population x(t). Thus, on a given closed interval [a, T], x(t) will be a close approximation to the actual population provided that $\delta >0$ is sufficiently small.

An initial value problem is usually considered well posed as a mathematical model for a real-world situation only if the differential equation has unique solutions that are continuous with respect to initial values. Otherwise it is unlikely that the initial value problem adequately mirrors the real-world situation.

An even stronger “continuous dependence” of solutions is often desirable. In addition to possible inaccuracy in the initial value, the function f(x, t) may not model precisely the physical situation. For instance, it may involve physical parameters (such as resistance coefficients) whose values cannot be measured with absolute precision. Birkhoff and Rota generalize the proof of Theorem 4 to establish the following result.

If $\mu >0$ is small, then (51) implies that the functions f and g appearing in the two differential equations, though different, are “close” to each other. If $ϵ>0$ is given, then it is apparent from (52) that

for all t in [a, T] if both $|\mathbf{\text{x}}(a)-\mathbf{\text{y}}(a)|$ and $\mu$ are sufficiently small. Thus Theorem 5 says (roughly) that if both the two initial values and the two differential equations in (50) are close to each other, then the two solutions remain close to each other for $a\leqq t\leqq T$.

For example, suppose that a falling body is subject both to constant gravitational acceleration g and to resistance proportional to some power of its velocity, so (with the positive axis directed downward) its velocity v satisfies the differential equation

Assume, however, that only an approximation $\overline{c}$ to the actual resistance c and an approximation $\overline{\rho }$ to the actual exponent $\rho$ are known. Then our mathematical model is based on the differential equation

instead of the actual equation in (54). Thus if we solve Eq. (55), we obtain only an approximation u(t) to the actual velocity v(t). But if the parameters $\overline{c}$ and $\overline{\rho }$ are sufficiently close to the actual values c and $\rho ,$ then the right-hand sides in (54) and (55) will be close to each other. If so, then Theorem 5 implies that the actual and approximate velocity functions v(t) and u(t) are close to each other. In this case, the approximation in (55) will be a good model of the actual physical situation.