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7.1. AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
function. As we’ll see, this may be too much to ask in some situations, but we will still be
able to make accurate approximations.
Preview Activity 7.1. The position of a moving object is given by the function s(t), where
s is measured in feet and t in seconds. We determine that the velocity is v(t) = 4t + 1 feet
per second.
(a) How much does the position change over the time interval [0, 4]?
(b) Does this give you enough information to determine s(4), the position at time
t = 4? If so, what is s(4)? If not, what additional information would you need to
know to determine s(4)?
(c) Suppose you are told that the object’s initial position s(0) = 7. Determine s(2), the
object’s position 2 seconds later.
(d) If you are told instead that the object’s initial position is s(0) = 3, what is s(2)?
(e) If we only know the velocity v(t) = 4t + 1, is it possible that the object’s position
at all times is s(t) = 2t 2 + t − 4? Explain how you know.
(f) Are there other possibilities for s(t)? If so, what are they?
(g) If, in addition to knowing the velocity function is v(t) = 4t + 1, we know the initial
position s(0), how many possibilities are there for s(t)?
⊲⊳
What is a differential equation?
A differential equation is an equation that describes the derivative, or derivatives, of a
function that is unknown to us. For instance, the equation
dy
dx
= x sin x
is a differential equation since it describes the derivative of a function y(x) that is unknown
to us.
As many important examples of differential equations involve quantities that change in
time, the independent variable in our discussion will frequently be time t. For instance, in
the preview activity, we considered the differential equation
ds
dt
= 4t + 1.
Knowing the velocity and the starting position of the object, we were able to find the
position at any later time.
7.1. AN INTRODUCTION TO DIFFERENTIAL EQUATIONS
function. As we’ll see, this may be too much to ask in some situations, but we will still be
able to make accurate approximations.
Preview Activity 7.1. The position of a moving object is given by the function s(t), where
s is measured in feet and t in seconds. We determine that the velocity is v(t) = 4t + 1 feet
per second.
(a) How much does the position change over the time interval [0, 4]?
(b) Does this give you enough information to determine s(4), the position at time
t = 4? If so, what is s(4)? If not, what additional information would you need to
know to determine s(4)?
(c) Suppose you are told that the object’s initial position s(0) = 7. Determine s(2), the
object’s position 2 seconds later.
(d) If you are told instead that the object’s initial position is s(0) = 3, what is s(2)?
(e) If we only know the velocity v(t) = 4t + 1, is it possible that the object’s position
at all times is s(t) = 2t 2 + t − 4? Explain how you know.
(f) Are there other possibilities for s(t)? If so, what are they?
(g) If, in addition to knowing the velocity function is v(t) = 4t + 1, we know the initial
position s(0), how many possibilities are there for s(t)?
⊲⊳
What is a differential equation?
A differential equation is an equation that describes the derivative, or derivatives, of a
function that is unknown to us. For instance, the equation
dy
dx
= x sin x
is a differential equation since it describes the derivative of a function y(x) that is unknown
to us.
As many important examples of differential equations involve quantities that change in
time, the independent variable in our discussion will frequently be time t. For instance, in
the preview activity, we considered the differential equation
ds
dt
= 4t + 1.
Knowing the velocity and the starting position of the object, we were able to find the
position at any later time.