  Ref.[BCOW17] introduced a pioneering quantum approach (coined BCOW algorithm)
for solving linear differential equations with optimal error tolerance.
Originally designed for a specific class of diagonalizable linear differential
equations, the algorithm was extended by Krovi in [Kro23] to encompass broader
classes, including non-diagonalizable and even singular matrices. Despite the
common misconception, the original algorithm is indeed applicable to
non-diagonalizable matrices, with diagonalisation primarily serving for
theoretical analyses to establish bounds on condition number and solution
error. By leveraging basic estimates from [Kro23], we derive bounds comparable
to those outlined in the Krovi algorithm, thereby reinstating the advantages of
the BCOW approach. Furthermore, we extend the BCOW algorithm to address
time-dependent linear differential equations by transforming non-autonomous
systems into higher-dimensional autonomous ones, a technique also applicable
for the Krovi algorithm.
