Chapter Review

In this chapter, we have considered two kinds of differential equations. The first kind expresses the derivative of an unknown function in terms of the dependent variable only, e.g.,

The second expresses the derivative as an explicit function of the independent variablealone, e.g.,

Every differential equation in this course will be of one or the other of these two types.

Either kind of differential equation can be expected to have infinitely many solutions. When we specify a single point on the solution curve (an initial condition), we usually single out a unique solution function. As this will always be true for the differential equations in this course, we simply assume that every initial value problem has a unique solution.

We recalled at the beginning of the chapter the useful visual device called the "slope field." The most important role of the slope field is as a conceptual tool. You should imagine each differential equation being represented by such a field — that is, by a large number of direction markers whose slopes are determined by the differential equation. Solving the differential equation then corresponds to finding curves (graphs of functions) that pass through the slope field always following the direction markers. And solving an initial value problem means picking out the particular solution curve that goes through the right initial point.

In the remaining sections, we studied applications that led naturally to the two different types of differential equation.

• Section 3.2 presented a scenario for application of Newton's Law of Cooling. The main step in determining the time of death of a murder victim was to find a solution of a differential equation in which the right-hand side involved only the unknown temperature function. Our technique for solution was to turn the problem into one we had already solved in Chapter 2.

• Section 3.3, in the context of a falling marble, we saw that we could find a formula for velocity from the assumption of constant acceleration, and then we could find a formula for distance from the formula for velocity. In each case, we were solving an initial value problem in which the right-hand side of the differential equation depended only on the independent variable, time.