DIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS
9

9.5
Linear Equations
DIFFERENTIAL EQUATIONS
In this section, we will learn:
How to solve linear equations
using an integrating factor.

LINEAR EQUATIONS
A first-order linear differential equation is
one that can be put into the form
where
P
and
Q
are continuous functions
on a given interval.
This type of equation occurs frequently in various
sciences, as we will see.
( )
( )
dy
P x y
Q x
dx
Equation 1

LINEAR EQUATIONS
An example of a linear equation is
xy’
+
y
= 2
x
because, for
x
≠ 0, it can be written
in the form
1
'
2
y
y
x
Equation 2

LINEAR EQUATIONS
Notice that this differential equation
is not separable.
It’s impossible to factor the expression for
y’
as a function of
x
times a function of
y
.

LINEAR EQUATIONS
However, we can still solve the equation
by noticing, by the Product Rule, that
xy’
+
y
= (
xy
)
’
So, we can rewrite the equation as:
(
xy
)
’
= 2
x

LINEAR EQUATIONS
If we now integrate both sides,
we get:
xy
=
x
2
+
C
or
y
=
x
+
C
/
x
If the differential equation had been in the form of
Equation 2, we would have had to initially multiply
each side of the equation by
x
.

INTEGRATING FACTOR
It turns out that every first-order linear
differential equation can be solved in a similar
fashion by multiplying both sides of Equation 1
by a suitable function
I
(
x
).
This is called an integrating factor.

LINEAR EQUATIONS
We try to find
I
so that the left side of
Equation 1, when multiplied by
I
(
x
), becomes
the derivative of the product
I
(
x
)
y
:
I
(
x
)(
y’
+
P
(
x
)
y
) = (
I
(
x
)
y
)
’
Equation 3

LINEAR EQUATIONS
If we can find such a function
I
, then
Equation 1 becomes:
(
I
(
x
)
y
)
’
=
I
(
x
)
Q
(
x
)
Integrating both sides, we would have:
I
(
x
)
y
=
∫
I
(
x
)
Q
(
x
)
dx
+
C

LINEAR EQUATIONS
So, the solution would be:
1
( )
( )
( )
( )
y x
I x Q x dx
C
I x
Equation 4

LINEAR EQUATIONS
To find such an
I
, we expand Equation 3 and
cancel terms:
I
(
x
)
y’
+
I
(
x
)
P
(
x
)
y
= (
I
(
x
)
y
)
’
=
I
’
(
x
)
y
+
I
(
x
)
y’
I
(
x
)
P
(
x
) =
I
’
(
x
)

SEPERABLE DIFFERENTIAL EQUATIONS
This is a separable differential equation for
I
,
which we solve as follows:
where
A
= ±
e
C
.

#### You've reached the end of your free preview.

Want to read all 52 pages?

- Fall '10
- capretta