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

##
*J. Das (n\'{e}e Chaudhuri) *

**Address.**

Department of Pure Mathematics, University of Calcutta, 35, Ballygunge Circular Road, Kolkata-700019, India

**E-mail. **jtdas2000@yahoo.co.in

**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
{\em 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 {\em 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.

**AMSclassification. **34B.

**Keywords. **Natural BC, compatible BCs with respect to a given DE.