condition, but the function must also be differentiable on an interval that contains the initial point.
Features such as vertical tangents or asymptotes restrict the domain of the solution. Therefore, even
when they are defined by the same algebraic representation, particular solutions with different initial
points may have different domains. Determining the proper domain is an important part of finding the
particular solution.
In §A of Chapter 8 we solved more differential equations involving motion along a straight line. In
§B we found parametric equations for the motion of a particle along a plane curve, given d.e.’s for the
components of its acceleration and velocity.
Rate of Change
A differential equation contains derivatives. A derivative gives information about the rate of change
of a function. For example:
(1) If P is the size of a population at time t, then we can interpret the d.e.
as saying that at any time t the rate at which the population is growing is proportional (3.25%) to its
size at that time.
(2) The d.e. tells us that at any time t the rate at which the quantity Q is decreasing
is proportional (0.0275%) to the quantity existing at that time.
(3) In psychology, one typical stimulus-response situation, known as logarithmic response, is that
in which the response y changes at a rate inversely proportional to the strength of the stimulus x. This
is expressed neatly by the differential equation
If we suppose, further, that there is no response when x = x 0 , then we have the condition y = 0 when x
= x 0.
(4) We are familiar with the d.e.
for the acceleration due to gravity acting on an object at a height s above ground level at time t. The
acceleration is the rate of change of the object’s velocity.
B. SLOPE FIELDS
In this section we solve differential equations by obtaining a slope field or calculator picture that
approximates the general solution. We call the graph of a solution of a d.e. a solution curve.
The slope field of a d.e. is based on the fact that the d.e. can be interpreted as a statement about
the slopes of its solution curves.
EXAMPLE 1
The d.e. tells us that at any point (x, y) on a solution curve the slope of the curve is equal to
its y-coordinate. Since the d.e. says that y is a function whose derivative is also y, we know thaty = ex