Steady-state oscillations in continuously inhomogeneous medium described by the generalized darboux equation

We refine a result due to L. V. Ovsyannikov on the general formof the second order linear differential equations with a nonzero generalized Laplace which are invariant admitting a Lie group of transformations of the maximal order with n > 2 independent variables for which the associated Riemannian spaces have nonzero curvature. We show that the set of these equations is exhausted by the generalized Darboux equation and the Ovsyannikov equation. We find the operators acting on the set of solutions in every one-parameter family of generalized Darboux equations. For the elliptic generalized Darboux equation possessing the maximal symmetry and describing steadystate oscillations in continuously inhomogeneous medium with a degeneration hyperplane, the group analysis methods yield the exact solutions to boundary value problems for some regions (a generalized Poisson formula) which in particular can be the test solutions in simulating steadystate oscillations in continuously inhomogeneous media. © 2010 Pleiades Publishing, Ltd.