Advanced Search

The ideas of the present paper have originated from the observation that all solutions of the linear homogeneous differential equation (DE) $y^{\text{'}\text{'}}\left(t\right)+y\left(t\right)=0$ satisfy the non-trivial linear homogeneous boundary conditions (BCs) $y\left(0\right)+y\left(\pi \right)=0$, $y^{\text{'}}\left(0\right)+y^{\text{'}}\left(\pi \right)=0$. 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...