On the boundary conditions associated with second-order linear homogeneous differential equations

Das, J.. "On the boundary conditions associated with second-order linear homogeneous differential equations." Archivum Mathematicum 040.3 (2004): 301-313. <http://eudml.org/doc/249307>.

@article{Das2004,
abstract = {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.},
author = {Das, J.},
journal = {Archivum Mathematicum},
keywords = {natural BC; compatible BCs with respect to a given DE; natural BC; compatible BCs with respect to a given DE},
language = {eng},
number = {3},
pages = {301-313},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On the boundary conditions associated with second-order linear homogeneous differential equations},
url = {http://eudml.org/doc/249307},
volume = {040},
year = {2004},
}

TY - JOUR
AU - Das, J.
TI - On the boundary conditions associated with second-order linear homogeneous differential equations
JO - Archivum Mathematicum
PY - 2004
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 040
IS - 3
SP - 301
EP - 313
AB - 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.
LA - eng
KW - natural BC; compatible BCs with respect to a given DE; natural BC; compatible BCs with respect to a given DE
UR - http://eudml.org/doc/249307
ER -