# Article

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Keywords:
natural BC; compatible BCs with respect to a given DE
Summary:
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.
References:
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[2] Ince E. L.: Ordinary Differential Equations. Dover, New York, 1956. MR 0010757
[3] Eastham M. S. P.: Theory of Ordinary Differential Equations. Van Nostrand Reinhold, London, 1970. Zbl 0195.37001

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