Math 334 Lecture #3
§
2.1: First Order Linear ODE’s
The first order ODE’s
dy
dt
=

1
5
y
+ 9
.
8
and
ty
= 4
t
2

2
y
are examples of the general first order linear ODE
dy
dt
+
p
(
t
)
y
=
g
(
t
)
where
p
(
t
) and
g
(
t
) are functions continuous on a common open interval
I
.
Method of Solution (Leibniz).
The expression on the lefthand side of the differen
tial equation,
dy
dt
+
p
(
t
)
y,
is similar to the derivative of the product of
y
with a differentiable function, say,
μ
(
t
):
d
dt
μ
(
t
)
y
=
μ
(
t
)
dy
dt
+
μ
(
t
)
y.
This suggests multiplying the ODE through by
μ
(
t
):
μ
(
t
)
dy
dt
+
μ
(
t
)
p
(
t
)
y
=
μ
(
t
)
g
(
t
)
.
For the lefthand side of this to be [
μ
(
t
)
y
]
requires that
μ
(
t
) satisfies the differential
equation
μ
(
t
)
p
(
t
) =
μ
(
t
)
.
[To solve one differential equation we have to solve another differential equation! We will
see this idea again and again.]
Rewrite the ODE for
μ
with all the
μ
’s on one side:
μ
(
t
)
μ
(
t
)
=
p
(
t
)
.
The lefthand side is the derivative of a natural log:
d
dt
ln

μ
(
t
)

=
p
(
t
).
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 Spring '11
 Smith
 Math, Derivative, dt, Leibniz, y. dt dt

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