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Unformatted text preview: 1 Solutions 99 . It follows that the Wronskian is zero for all
.
3. The condition that the coeﬃcient function
be nonzero in Theorem 4 and Proposition 6 is essential. Here the coeﬃcient function, ,
of
is zero at
, so Proposition 6 does not apply on
.
The largest open intervals on which
is nonzero are
and
. On each of these intervals
and
are linearly dependent.
4. Consider the cases
and
. The veriﬁcation is then straightforward.
5. Again the condition that the coeﬃcient function
be nonzero is
essential. The Uniqueness and Existence theorem does not apply. Section 4.3
1. The indicial polynomial is
two distinct roots and
. The fundamental set is
solution is
2. The indicial polynomial is
two distinct roots . There are
. The general
. There are and . The fundamental set is . The general solution is
3. The indicial polynomial is
root, . There is one , with multiplicity . The fundamental set is . The general solution is
. There are 4. The indicial polynomial is
two distinct roots and . The fundamental set is . The general solution is
5. The indicial polynomial is . The root is with multiplicity . The fundamental set is . The general solution is
6. The indicial polynomial is
are and
. The fundamental set is . The roots
. The general solution is 8. The indicial polynomial is
. There are two complex roots,
The fundamental set is and . . The general solution is ...
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 Fall '08
 BELL,D

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