### 8.2.1. 1^{st} Order LDE

A linear differential equation (LDE) of 1^{st}
order takes the form

y
′
+P(
x
)y=Q(
x
)
(
y
′
=
dy
dx
)
(8.1)

where *P*
and *Q* are known functions and *y* is the unknown. This is one of the few type of DEs that can
be solved by elementary means, i.e., through algebraic operations,
compositions, and integrations.

#### Theorem 8.1.

Let *P*
and *Q* be continuous on an open
interval *J*. Then there exists 1 and only 1 function
y=f(
x
)
that satisfies both (8.1) and the initial
condition
f(
a
)=b
,
where
a∈J
and
b∈R
. Furthermore,

f(
x
)=b
e
−A(
x
)
+
e
−A(
x
)
∫
a
x
dt
Q(
t
)
e
A(
t
)
(8.2)

where
A(
x
)=
∫
a
x
dt
P(
t
)
.

##### Proof

Using
A
′
(
x
)=P(
x
)
,
we see that
e
A(
x
)
×(
8.1
)
gives

e
A
y
′
+
e
A
A
′
y=
e
A
Q

Þ
(
e
A
y
)
′
=
e
A
Q
(8.1a)

The factor
e
A(
x
)
is called an **integrating factor** since we can now integrate (8.1a) over the interval
[
a,x
]
to get

e
A(
x
)
f(
x
)−
e
A(
a
)
f(
a
)=
∫
a
x
dt
Q(
t
)
e
A(
t
)

Using
A(
a
)=
∫
a
a
dt
P(
t
)=0
and
f(
a
)=b
,
eq(8.2) is recovered. Uniqueness of this
solution is guaranteed by Theorem 8.3 of §8.5.