17
Differential Equations
Many physical phenomena can be modeled using the language of calculus. For example,
observational evidence suggests that the temperature of a cup of tea (or some other liquid)
in a room of constant temperature will cool over time at a rate proportional to the difference
between the room temperature and the temperature of the tea.
In symbols, if
t
is the time,
M
is the room temperature, and
f
(
t
) is the temperature
of the tea at time
t
then
f
′
(
t
) =
k
(
M
−
f
(
t
)) where
k >
0 is a constant which will depend
on the kind of tea (or more generally the kind of liquid) but not on the room temperature
or the temperature of the tea. This is
Newton’s law of cooling
and the equation that
we just wrote down is an example of a
differential equation
. Ideally we would like to
solve this equation, namely, find the function
f
(
t
) that describes the temperature over
time, though this often turns out to be impossible, in which case various approximation
techniques must be used. The use and solution of differential equations is an important
field of mathematics; here we see how to solve some simple but useful types of differential
equation.
Informally, a differential equation is an equation in which one or more of the derivatives
of some function appear. Typically, a scientific theory will produce a differential equation
(or a system of differential equations) that describes or governs some physical process, but
the theory will not produce the desired function or functions directly.
Recall from section 6.2 that when the variable is time the derivative of a function
y
(
t
)
is sometimes written as ˙
y
instead of
y
′
; this is quite common in the study of differential
equations.
425
426
Chapter 17 Differential Equations
We start by considering equations in which only the first derivative of the function appears.
DEFINITION 17.1
A
first order differential equation
is an equation of the form
F
(
t, y,
˙
y
) = 0. A solution of a first order differential equation is a function
f
(
t
) that makes
F
(
t, f
(
t
)
, f
′
(
t
)) = 0 for every value of
t
.
Here,
F
is a function of three variables which we label
t
,
y
, and ˙
y
. It is understood
that ˙
y
will explicitly appear in the equation although
t
and
y
need not. The term “first
order” means that the first derivative of
y
appears, but no higher order derivatives do.
EXAMPLE 17.2
The equation from Newton’s law of cooling, ˙
y
=
k
(
M
−
y
) is a first
order differential equation;
F
(
t, y,
˙
y
) =
k
(
M
−
y
)
−
˙
y
.
EXAMPLE 17.3
˙
y
=
t
2
+1 is a first order differential equation;
F
(
t, y,
˙
y
) = ˙
y
−
t
2
−
1.
All solutions to this equation are of the form
t
3
/
3 +
t
+
C
.
DEFINITION 17.4
A
first order initial value problem
is a system of equations of
the form
F
(
t, y,
˙
y
) = 0,
y
(
t
0
) =
y
0
. Here
t
0
is a fixed time and
y
0
is a number. A solution
of an initial value problem is a solution
f
(
t
) of the differential equation that also satisfies
the
initial condition
f
(
t
0
) =
y
0
.
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 Fall '11
 GUICHARD
 Equations, Derivative, Elementary algebra

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