UNDETERMINED COEFFICIENTS
for FIRST ORDER LINEAR EQUATIONS
This method is useful for solving nonhomogeneous linear equations written in the form
dy
dx
+
k y
=
g
(
x
)
,
where
k
is a nonzero constant and
g
is
1. a polynomial,
2. an exponential
e
rt
,
3. a product of an exponential and a polynomial,
4. a sum of trigonometric functions sin (
ω t
), cos (
ω t
),
5. a sum of products
e
rt
sin (
ω t
)
, e
rt
cos (
ω t
),
6. a sum of terms
p
(
t
)
,
sin (
ω t
) +
q
(
t
) cos (
ω t
), where
p
and
q
are polynomials.
Here are a couple more examples.
Example 1:
Find the general solution of
y

4
y
= 8
x
2
.
Here we take a trial solution to be a general polynomial of degree two
y
p
(
x
) =
A x
2
+
B x
+
C .
Then
y
p
(
x
) = 2
Ax
+
B
and substituting we have
(2
A x
+
B
)

4 (
A x
2
+
B x
+
C
) = 8
x
2
.
Now, collecting like powers of
x
we rewrite this equation as

4
A x
2
+ (2
A

4
B
)
x
+ (
B

4
C
) = 8
x
2
,
and comparing coefficients of like terms on both sides of the equation gives

4
A
=
8
,
2
A

4
B
= 0, and
B

4
C
= 0, from which we see that
A
=

2
, B
=

1, and
C
=

1
/
4. Hence
y
p
(
x
) =

2
x
2

x

1
4
,
and the general solution is
1
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y
(
x
) =
C e
4
x
+

2
x
2

x

1
4
.
Example 2:
Now consider the equation
y

4
y
= 2
x
cos (
x
). To find a particular solution, we take a
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 Spring '08
 Staff
 Linear Equations, Equations, Cos, general solution

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