We're looking for a solution
which has second derivative
Substituting these into the ODE gives
Right away we see , and the coefficients are given according to the recurrence
There's a dependency between terms in the sequence that are 4 indices apart, so we consider 4 different cases.
- If , where is an integer, then
and so on, with the general pattern
- If , then
and so on, with
- If or , then
Then the solution to this ODE is