Supplementary material on finding particular solutions to systems of linear first order
equations
Diagonalization Method:
Consider the system of n linear first order constant coefficient equations
(
29
y
Ay
g t
′ =
+
.
If
A
has n linearly independent eigenvectors
(
29
(
29
(
29
1
1
,
,
,
n
x
x
x
×××
then form the matrix
(
29
(
29
(
29
1
1
,
,
,
n
X
x
x
x
=
×××
where each column is one of the eigenvectors.
Making the
substitution
y
Xu
=
we get
(
29
(
29
(
29
(
29
1
1
1
1
2
1
0
0
0
0
0
0
n
Xu
AXu
g t
X
Xu
X
AX u
X
g t
u
u
X
g t
λ
λ
λ




′
′
=
+
⇒
=
+
×××
÷
×××
÷
′
⇒
=
+
÷
×
×
×
÷
×××
Hence we now have n uncoupled linear first order equations that can be solved by the
methods learn in the first differential equations course.
Once we have the
u’s
simply take
y
Xu
=
to get the particular solution.
Method of Variation of Parameters:
Consider the system of n linear first order equations of the form
(
29
(
29
y
A t
y
g t
′ =
+
.
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 Fall '08
 MEI
 Xu, yu, Fundamental matrix

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