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Fall 2003 Math 308/501–502
6 HigherOrder Linear Differential Eqs
6.4/4.6 Variation of Parameters
Mon, 06/Oct
c
±
2003, Art Belmonte
Summary
Let
L
[
y
]
=
y
(
n
)
(
x
)
+
n
X
j
=
1
p
j
(
x
)
y
(
n

j
)
(
x
)
=
g
(
x
)
be a
nonhomogeneous
n
thorder linear differential equation in
standard linear form
(SLF). Recall that if
y
p
is a particular
solution of this nonhomogeneous equation and
y
h
is a general
solution to the associated homogeneous equation
L
[
y
]
=
0, then
a general solution of the nonhomogeneous equation is given by
y
=
y
p
+
y
h
.
Variation of Parameters (VOP)
The following method produces a
general solution
of the
nonhomogeneous equation.
First put the differential equation in
standard linear form!
1. Obtain a fundamental set of solutions
y
f
=
[
y
1
,...,
y
n
], a
row vector, of
L
[
y
]
=
0. Given a column vector of constants,
c
=
[
c
1
;
...
;
c
n
], form a general solution of this
homogeneous equation,
y
h
=
y
f
c
=
n
X
k
=
1
c
k
y
k
.
2. Form the column vector
b
=
[0
;
0
;
;
0
;
g
], all of whose
entries are zeros except the last one, which is the nonhomo
geneity
g
(righthand side of the nonhomogeneous equation).
Compute the Wronskian matrix
M
of
y
f
.So
lve
Mv
p
=
b
for
v
p
(
v
prime), a column vector:
vp = M
\
b
.
3. Integrate
v
p
to obtain
v
, a column vector. (Don’t worry about
constants of integration.)
4. A
particular solution
of the nonhomogeneous equation is
y
p
=
y
f
v
.
5. A
general solution
of the nonhomogeneous equation is
y
=
y
p
+
y
h
.
Use MATLAB for your computations!
The operations involved in the variation or parameters procedure
are tailormade for MATLAB: matrix multiplication, solving
linear systems, vector integration, etc.
MATLAB Examples
To facilitate computation of the Wronskian matrix, I wrote a
function Mﬁle named
wron
. Type “
help wron
” at a MATLAB
prompt to learn about it. If you’d like to see the (short) code, type
“
type wron
” at a MATLAB prompt.
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 Spring '08
 comech
 matlab

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