1)
Let
be a fundamental matrix for the problem
y
Ay
that has the property
that
(0)
.
I
Show that
1
( ) ( )
Z
t
s
and
2
()
Z
t
s
both satisfy the
initial value problem
, (0, )
( )
Z
AZ Z
s
s
.
Solution:
Since
t
is a fundamental matrix for
y
Ay
we have
( )
( )
( ) ( )
( ) ( )
( ( ) ( ))
( ) ( ).
t
A
t
t
s
A
t
s
t
s
A
t
s
Therefore
( ) ( )
ts
satisfies the equation.
We also have
(0) ( )
( )
( )
s
I
s
s
For the initial condition.
Since
t
is a fundamental matrix for
y
Ay
we have
( ).
d
A
d
Let
(
).
d
t
s
t
s
A
t
s
dt
Therefore
satisfies the equation.
We also have
0
(
)
( )
t
t
s
s
so exactly the same initial condition is satisfied. Since linear initial value problems
have unique solutions and we find that
and
( ) ( )
satisfy the same initial
value problem we can conclude that they are equal.
2)
Find the general solution to
t
t
e
e
y
y
2
2
4
1
1
2
by the method of variation of parameters as described in our text and by the
method of variation of parameters using the special fundamental matrix
as
described in class.
Solution:
A fundamental matrix for the homogeneous equations is easily found and is
t
t
t
t
e
e
e
e
Y
2
3
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 Fall '08
 MEI
 Cos, 1g, Xu, Fundamental matrix

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