MA366
Sathaye
Fast methods for some chapter 7 problems.
We show some faster methods to quickly get a fundamental matrix for linear systems of differential
equations.
Two variable systems.
1.
Basic formulas.
We consider a system
X
0
=
AX
where
A
is a 2
×
2 matrix with constant
coefficients.
Let
A
=
a
b
c
d
.
We note that its characteristic polynomial has the formula
P
A
(
λ
) =
λ
2
-
λ
(
a
+
d
) +
ad
-
bc
.
We typically solve for eigenvalues by finding the roots of
P
A
(
λ
) and depending on the nature of
roots, we have different techniques for solving the system.
2.
Fundamental Matrix
For this discussion, we may even consider an
n
×
n
matrix
A
giving a system of
n
equations in
n
variables. A Fundamental matrix for the system
X
0
=
AX
is defined to be an
n
×
n
matrix Ψ(
t
)
whose columns are
n
independent solutions of the system.
This can be shown equivalent to the condition Ψ
0
(
t
) =
A
Ψ(
t
) together with the condition that
Ψ(
t
) is invertible on an interval where the solutions are discussed.
The Special Fundamental Matrix is given by
Φ(
t
) =
exp
(
At
) =
I
+
At
+
A
2
t
2
/
2! +
· · ·
Its columns are independent solutions of the system and have the added property that Φ(0) =
I
.
Indeed, this extra condition identifies a Fundamental Matrix as a Special Fundamental Matrix.
A general solution to the system
X
0
=
AX
is then given by
X
(
t
) = Φ(
t
)
C
where
C
is a column
of arbitrary constants. In particular,
X
= Φ(
t
)
X
(0)
where
X
(0) =
X
(0)
is the given set of initial
conditions.
We give direct methods to find this matrix Φ(
t
).
3.
Exponential shift.
Consider a substitution
X
= exp(
rt
)
Y
. Substitution in the equation
X
0
=
AX
gives
X
0
= exp(
rt
)
rY
+ exp(
rt
)
Y
0
=
A
exp(
rt
)
Y.
By canceling exp(
rt
) and rearranging, we see that
Y
0
= (
A
-
rI
)
Y.
It is easy to see that the characteristic polynomial of
A
-
rI
is simply
P
A
(
λ
+
r
). We shall use
this substitution to simplify our characteristic polynomial.
Finally, whatever special fundamental matrix is found for the system
Y
0
= (
a
-
rI
)
Y
, we may
multiply it by exp(
rt
) to get the matrix for the system
X
0
=
AX
.
1
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4.
The Cayley-Hamilton Theorem.
Suppose
A
is an
n
×
n
matrix with characteristic polynomial
P
A
(
λ
) = (
-
1)
n
λ
n
+
a
1
λ
(
n
-
1)
+
· · ·
+
a
(
n
-
1)
λ
+
a
n
then
P
A
(
A
) = (
-
1)
n
A
n
+
a
1
A
(
n
-
1)
+
· · ·
+
a
(
n
-
1)
A
+
a
n
I
= 0
.
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- Fall '09
- EdrayGoins
- Differential Equations, Equations, Linear Systems, Formulas, Elementary algebra, Characteristic polynomial, Partial differential equation, special fundamental matrix
-
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