Unformatted text preview: olution may be written as
x 7. Solution of the ODE requires analysis of the algebraic equations
.
For a nonzero solution, we must have
A
I
. The only root
is
, which is an eigenvalue of multiplicity two. Substituting
into the
coefficient matrix, the system reduces to the single equation
. Hence the
corresponding eigenvector is
One solution is
x . For a second linearly independent solution, we search for a generalized eigenvector.
Its components satisfy ________________________________________________________________________
page 416 ————————————————————————— CHAPTER 7. —— ,
that is,
. Let
, some arbitrary constant. Then
It follows that a second solution is given by
x Dropping the last term, the general solution is
x
Imposing the initial conditions, we require that ,
which results in and Therefore the solution of the IVP is
x ________________________________________________________________________
page 417 —————————————————————————— CHAPTER 7. —...
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 Spring '08
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 Linear Algebra, eigenvector

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