If a sequence is defined by
a recurrence relation of the form
where
and
are constants we may assume a solution of the form
We
obtain a quadratic equation, which we solve. Each solution of the
quadratic gives rise to a solution of the original problem, the
general solution being obtained by adding these two solutions. Each
solution gives rise to an arbitrary constant. We can find these
arbitrary constants using initial (or boundary) conditions.
whereand
are constants we may assume a solution of the formWe
obtain a quadratic equation, which we solve. Each solution of the
quadratic gives rise to a solution of the original problem, the
general solution being obtained by adding these two solutions. Each
solution gives rise to an arbitrary constant. We can find these
arbitrary constants using initial (or boundary) conditions.
Suppose for example that we
have
with
and
withand
Assume a solution of the
form
then
and
Substitute these into the recurrence relation to obtain
thenand
Substitute these into the recurrence relation to obtain
Divide throughout by
to
obtain
to
obtain
Hence
or
and
orand
To find
and
use
the initial conditionsand
anduse
the initial conditionsand
(1)
(2)
(2)– (1) gives
then
sub this into (1) to obtain
then
sub this into (1) to obtain
The solution is
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