MATH2860U:
Chapter 4 cont…
1
HIGHERORDER DIFFERENTIAL
EQUATIONS cont…
Reduction of Order (Section 4.2 of Zill and Cullen, pg. 130)
Recall:
We’ve just spent time studying solutions to linear 2
nd
order differential
equations, and found that their solution could be expressed as
2
2
1
1
y
c
y
c
y
where
1
y
and
2
y
are linearly independent solutions.
Let’s explore this by considering a simple example:
Suppose that we know that
x
e
x
y
3
1
)
(
is a solution to
0
9
6
y
y
y
…what is a 2
nd
linearly independent
solution?
This technique used in the above example is known as the method of reduction of order.
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Chapter 4 cont…
2
Question:
What did we just do, and why did it work?
Question:
Will we have to go through this lengthy process each time?
Theorem:
For a 2
nd
order differential equation in standard form
0
)
(
)
(
y
x
Q
y
x
P
y
,
if one solution
)
(
1
x
y
is known, then
reduction of order
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 Spring '10
 Kidnan
 Derivative, Quadratic equation, Elementary algebra, Cullen

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