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

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

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

E-mail. jtdas2000@yahoo.co.in

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.