The ideas of the present paper have originated from the observation that all solutions of the linear homogeneous differential equation (DE) $y^{\prime \prime }(t) + y(t)=0$ satisfy the non-trivial linear homogeneous boundary conditions (BCs) $y(0) + y(\pi )=0$, $y^{\prime }(0) + y^{\prime }(\pi )=0$. Such a BC is referred to as a natural BC (NBC) with respect to the given DE, considered on the interval $[0, \pi ]$. This observation suggests the following queries : (i) Will each second-order linear homogeneous DE possess a natural BC ? (ii) How many linearly independent natural BCs can a DE possess ? The present paper answers these queries. It also establishes that any non-trivial homogeneous mixed BC, which is not a NBC with respect to the given linear homogeneous DE, determines uniquely (up to a constant multiplier), the solution of the DE. Two BCs are said to be compatible with respect to a given DE if both of them determine the same solution of the DE. Conditions for the compatibility of sets of two and three BCs with respect to a given DE have also been determined.
@article{107913,
author = {J. Das},
title = {On the boundary conditions associated with second-order linear homogeneous differential equations},
journal = {Archivum Mathematicum},
volume = {040},
year = {2004},
pages = {301-313},
zbl = {1117.34008},
mrnumber = {2107026},
language = {en},
url = {http://dml.mathdoc.fr/item/107913}
}
Das, J. On the boundary conditions associated with second-order linear homogeneous differential equations. Archivum Mathematicum, Tome 040 (2004) pp. 301-313. http://gdmltest.u-ga.fr/item/107913/
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