On the Solutions of a Second Order Differential Equation Arising in the Theory of Resonant Oscillations in a Tank

The properties of the solutions of the differential equation y″ = y² ? x² ? c subject to the condition that y is bounded for all finite x discussed. The arguments of Holmes and Spence have been used by Ockendon, Ockendon, and Johnson to show that there are no solutions if c is large and negative. Numerically we find that solutions exist provided c is greater than a critical value c* and estimate this value to be c* = ?…. As x tends to + ∞ the solutions are asymptotic to . The relation between A₊ and ?₊ are found analytically as A₊ → ∞. This problem arises as a connection problem in the theory of resonant oscillations of water waves.