MATH 150
Resonance
Cheung Man Wai
Tutorial 7
1
When inhomogeneous term is a solution of the
homogeneous solution
Notice that in the examples in our previous tutorial, we have
g
(
t
) =
x
h
(
t
).
What will happen if
g
(
t
) =
x
h
(
t
)? The strategy is to multiply
t
to your guessed
x
p
(
t
).
Example 1
Find a particular solution of
¨
x

3 ˙
x

4
x
= 2
e

t
.
Remark
.
Proceeding as in the lecture note, we assume that
x
(
t
) =
Ae

t
.
By
substituting in the question, we then obtain
(
A
+ 3
A

4
A
)
e

t
= 2
e

t
.
Since the lefthandside of above vanishes, there is no choice of
A
that satisfies
this equation. Thus there is no particular solution of the question of the assumed
form. The reason for this possibly unexpected result becomes clear if we solve
the homogeneous equation
¨
x

3 ˙
x

4
x
= 0
that corresponds to the question. A fundamental set of solution of homogeneous
part is
x
1
(
t
) =
e

t
and
x
2
=
e
4
t
.
Thus our assumed particular solution is actu
ally a solution of the homogeneous equation. Consequently, it cannot possibly
be a solution of the inhomogeneous equation. To find a particular solution, we
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 Spring '09
 T.Qian
 Math, Quadratic equation, Elementary algebra, Homogeneity

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