Reduction of order  Wikipedia, the free encyclopedia
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4/2/08 6:40 AM
Reduction of order
From Wikipedia, the free encyclopedia
Reduction of order
is a technique in mathematics for solving secondorder ordinary differential equations. It is employed when one
solution
y
1
(
x
) is known and a second linearly independent solution
y
2
(
x
) is desired.
A Simple Example
Consider the general secondorder constant coefficient ODE
where
a
,
b
,
c
are real nonzero coefficients. Furthermore, assume that the associated characteristic equation
has repeated roots (i.e. the discriminant,
b
2
 4
ac
, vanishes). Thus we have
Thus our one solution to the ODE is
To find a second solution we take as an ansatz
where
v
(
x
) is an unknown function to be determined. Since
y
2
(
x
) must satisfy the original ODE, we substitute it back in to get
Rearranging this equation in terms of the derivatives of
v
(
x
) we get
Since we know that
y
1
(
x
) is a solution to the original problem, the coefficient of the last term is equal to zero. Furthermore,
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 Winter '08
 JaberAbdualrahman
 Math, Vector Space, Wikipedia, ORDINARY DIFFERENTIAL EQUATIONS, single solution, linearly independent solution

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