Unformatted text preview: ———————————————— CHAPTER 7. ——
x
be any solution of the homogeneous equation. That is, x
P x . We know
that x x , in which x is a linear combination of some fundamental solution. By
linearity of the differential equation, it follows that x
x
x is a solution of the
ODE. Based on the uniqueness theorem, all solutions must have this form.
7 . By definition,
x x . The Wronskian vanishes at
independent on
. It follows from Theorem
must be discontinuous at and . Hence the vectors are linearly that one or more of the coefficients of the ODE
and
. If not, the Wronskian would not vanish. . Let
x
Then
x
On the other hand,
x Comparing coefficients, we find that Solution of this system of equations results in Hence the vectors are solutions of the ODE ________________________________________________________________________
page 363 —————————————————————————— CHAPTER 7. —— x x 8. Suppose that the solutions x , x , , x are linearly dependent at
there are constants
not all...
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 Spring '08
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 Linear Algebra, eigenvector

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