  The $so(5)$ (i.e., $B_2$) quantum integrable spin chains with both periodic
and non-diagonal boundaries are studied via the off-diagonal Bethe Ansatz
method. By using the fusion technique, sufficient operator product identities
(comparing to those in [1]) to determine the spectrum of the transfer matrices
are derived. For the periodic case, we recover the results obtained in
\cite{NYReshetikhin1}, while for the non-diagonal boundary case, a new
inhomogeneous $T-Q$ relation is constructed. The present method can be directly
generalized to deal with the $so(2n+1)$ (i.e., $B_n$) quantum integrable spin
chains with general boundaries.
