MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16
Second Order Differential Equations
Linear Equations.
We briefly recall some facts about differential equations which we have seen over the
past few weeks. Recall that we use the notation
y
(
n
)
=
d
n
y
dt
n
to denote the
n
th derivative. We say that an equation of the form
F
t, y, y
(1)
, . . . , y
(
n
)
= 0
is an
n
th order differential equation
. In particular, we can express the highest order derivative in terms of
the lower order derivatives:
d
n
y
dt
n
=
G
t, y, y
(1)
, . . . , y
(
n

1)
for some function
G
(
t, y
1
, y
2
, . . . , y
n
). When
n
= 2, we call an equation of the form
d
2
y
dt
2
=
G
t, y,
dy
dt
a
second order differential equation
.
Recall that a first order differential equation is said to be a linear equation if it is in the form
dy
dt
=
G
(
t, y
)
where
G
(
t, y
) =
g
(
t
)

p
(
t
)
y
for some functions
p
(
t
) and
g
(
t
).
Similarly, we say that a second order differential equation is a
linear
equation
if it is in the form
d
2
y
dt
2
=
G
t, y,
dy
dt
where
G
(
t, y
1
, y
2
) =
g
(
t
)

q
(
t
)
y

p
(
t
)
dy
dt
for some functions
p
(
t
),
q
(
t
), and
g
(
t
). Any second order differential equation which cannot be placed in this
form is called a
nonlinear equation
. Equivalently, we say that a second order differential equation is linear if
it is in the form
d
2
y
dt
2
+
p
(
t
)
dy
dt
+
q
(
t
)
y
=
g
(
t
)
.
We mention in passing that sometimes we consider second order differential equations in the form
a
(
t
)
y
00
+
b
(
t
)
y
0
+
c
(
t
)
y
=
f
(
t
)
.
Upon dividing both sides by
a
(
t
), we can place this equation in the one above, where
p
(
t
) =
b
(
t
)
a
(
t
)
,
q
(
t
) =
c
(
t
)
a
(
t
)
,
and
g
(
t
) =
f
(
t
)
a
(
t
)
.
Example.
Consider an object of mass
m
which is under the influence of gravity and air resistance.
Let
x
=
x
(
t
) denote the height of the mass at time
t
. The force due to gravity is
m g
, and the force due to air
resistance is
γ v
for some constant
γ
. Newton’s Second Law of Motion states that