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models equations. described above.
di erential the situationAy y(1) = y0 where
4. Solve the the IVP y = The, relationship for arbitrary functions is more
subtle. To see how, consider the followingsystem, if0any.
example
(b) Find the equilibrium solutions 2x the , if x
of
14
2 cos(x) 3 .
A = y solution a they0
(c) Find the general 1 (x) = ofnd 2x =
system.
47
2
x
, if x 0
2cos(xthe , if x < 0 two problems. If you
cos(x) following
* You may use Maple to help you with )
5. Consider the DE y y1 (x)y=
= A where x
your worksheet(s) if x <your assignment.
, with 0
5 .do, include a printout of 0
5
3x
A=
. cos(x) , x IR
y ( x) y
8. Consider the system0y 2= 5 =, where x)
A
3 cos(
3x multiplicity 2
1
(a) Use the deﬁnitionytoeigenvalue4of 2 (x),,yx(x)IR that has two linearly
2 )=
(a) Show that A has one(xshow that yx)
are linearly dependent
3 cos(41 3 the2general solution of the DE.
A Hence
independent eigenvectors. = 0 give .
on x 0.
0 0 (4
(a) Use the deﬁnition to show that y1solve 2 (x) resulting equations and
(b) Write the DE in component form,), x), y) are are linearly dependent
(b) Use the deﬁnition show that y1 (x y2 (x the linearly dependent on
on x that A...
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 Fall '11
 Noidea
 Linear Algebra, Boundary value problem, Eigenvalue, eigenvector and eigenspace, Generalized eigenvector, di erential equations, andxgeneralized eigenvector u. deﬁnition

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