First Order Initial Value Problem

(2)

An initial value problem usually has just one solution, which will be a particular solution of the differential equation. This can be seen intuitively as follows. Figure 14.1.3 shows a moving point controlled by an infinitesimal driver with a steering wheel. At t = t₀, the moving point starts at (t₀, y₀). At each t > t₀, the infinitesimal driver measures his position (t, y), computes the value off(t, y), and turns the steering wheel so that the slope will be f(t, y). The curve traced out by this point will be the solution of the initial value problem.

Figure 14.1.3

The general problem of solving a differential equation is difficult. In Section 8.6, we gave a method for solving differential equations of an especially simple type, called equations with separable variables. As a starting point in this chapter, we take another look at these differential equations.