Barrons AP Calculus - David Bock

CHAPTER 9 Differential Equations

Concepts and Skills

In this chapter, we review how to write and solve differential equations, specifically,

• writing differential equations to model dynamic situations;


• understanding a slope field as a graphical representation of a differential equation and its
  solutions;


• finding general and particular solutions of separable differential equations;


• and using differential equations to analyze growth and decay.
  We also review two additional BC Calculus topics:


• Euler’s method to estimate numerical solutions


• and using differential equations to analyze logistic growth and decay.

A. BASIC DEFINITIONS

Differential equation

A differential equation (d.e.) is any equation involving a derivative. In §E of Chapter 5 we solved
some simple differential equations. In Example 50, we were given the velocity at time t of a particle
moving along the x-axis:

From this we found the antiderivative:

If the initial position (at time t = 0) of the particle is x = 3, then
x(0) = 0 − 0 + C = 3,
and C = 3. So the solution to the initial-value problem is

A solution of a d.e. is any function that satisfies it. We see from (2) above that the d.e. (1) has an
infinite number of solutions—one for each real value of C. We call the family of functions (2) the
general solution of the d.e. (1). With the given initial condition x(0) = 3, we determined C, thus
finding the unique solution (3). This is called the particular solution.
Note that the particular solution must not only satisfy the differential equation and the initial