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 investigate 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