Geodesic
On the boundary conditions associated with second-order linear homogeneous differential equations
Archivum mathematicum, Tome 40 (2004) no. 3, pp. 301-313

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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.
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.
Classification : 34B05, 34B24
Keywords: natural BC; compatible BCs with respect to a given DE 
Das, J. On the boundary conditions associated with second-order linear homogeneous differential equations. Archivum mathematicum, Tome 40 (2004) no. 3, pp. 301-313. http://geodesic.mathdoc.fr/item/ARM_2004_40_3_a10/
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