Comparing approximate solutions with the slope field is encouraged. In Exercises 14-17, the appropriate step size must be determined via experimentation. Exercises 18 and 19 are applications of Euler's method to root finding. Lab 1.4 is very similar. In this section we present the Existence-Uniqueness Theorem.
Existence is stated and then taken for granted.
Uniqueness, however, is emphasized as a very useful tool for the qualitative study of solutions. We also deal with questions of the domain of definition for solutions. The exceptional student will have wondered about domains of definition in Section 1.2 where restricted domains are the norm. We take the dynamical systems point of view that a solution that escapes to infinity or that encounters a singular point of the differential equation at a finite time cannot be extended beyond that time. The information thus gained is qualitative.