1
Prof.
S.L. Phoenix
10/16/09
Math 2930
Method of Variation of Parameters for Higher Order Linear ODE’s
(Alternate derivation of Eq 33 in E&P Section 3.5)
Suppose we have the equation
11
nn
L
yy
p
x
y
px
y
p
x
y
g
x
(
1
)
which is eqn. (2) in E&P, except we use
gx
instead of
f
x
.
From the homogeneous equation
0
Ly
we can get
n
linearly independent complementary solutions
12
,
,...,
n
yx yx
yx
.
To get
a
particular
solution,
()
Y x
, we develop a general useful formula based on the method of
variation of parameters.
This method begins with the form
2 2
Yx uxy x u xy x
u xy x
(
2
)
where
,,
,
n
uxux
ux
arbitrary, but so far unknown functions.
At this stage, this form is
general enough to cover all possibilities for the particular solution.
In fact it looks like
overkill
, but as
we shall see, we will eventually apply
1
n
constraints.
To be a particular solution we must get the form (2) to satisfy the differential equation (1).
This means
we will need to take
n
derivatives of
Yx
, which through repeated applications of the chain rule will
cause a big mess unless we make some clever simplifications that we can get to work out in the end.
For the first derivative we use the chain rule of differentiation to give us
Yx uxyx uxyx
uxyx
uxyx uxyx
(
3
)
but we simplify this by setting the last half of the sum equal to zero, i.e., we apply the constraint
0
(
4
)
Thus we now only have
(
5
)
Taking a derivative of (5) we get
Yx ux
yx ux
(
6
)
but again we suppress the second part of the sum and apply the constraint
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11
2 2
0
nn
uxyx uxyx
uxyx
(
7
)
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 TERRELL,R
 Math, Differential Equations, Equations, Derivative, E&P, yn yn

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