Central Forces 2023
For each of the problems below, suppose you have been solving a differential equation using power series methods around the indicated point and you have derived the indicated recurrence relation. Write out the first five nonzero terms in the power series expansion. If the recurrence relation allows two solutions, write out the first four nonzero terms in each such solution.
In an expansion around the point \(z=1\), the recurrence relation is:
\[a_{n+1}=\frac{1}{n+1}\, a_n\]
In an expansion around the point \(z=0\), the recurrence relation is:
\[a_{n+2}=-\frac{(5-n)(6+n)}{(n+2)(n+1)}\, a_n\]