Caroline and Ron have 11 goldfish and all Caroline has three more than Ron how many fish do they each have
2 answers:
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We want a solution in the form
with derivatives
Substituting and its derivatives into the ODE,
gives
Shift the index on the second sum to have it start at :
and take the first term out of the other two sums. Then we can consolidate the sums into one that starts at :
and so the coefficients in the series solution are given by the recurrence,
or more simply, for ,
Note the dependency between every other coefficient. Consider the two cases,
- If
, where
is an integer, then
and so on, with the general pattern
- If
, then
and we would see that for all
.
So we have
so that one solution is
and the other is
I've attached a plot of the exact and series solutions below with ,
, and
to demonstrate that the series solution converges to the exact one.
