Abstract. It is shown that a pre-existing electron density ripple in a dense plasma can excite electron neutrino oscillations. For our purposes, we use the dispersion relation for neutrino oscillations and derive the Mathieu equation for the propagation of neutrino oscillations in the presence of a spatially oscillating electron density ripple. The Mathieu equation predicts instability of neutrino oscillations. The criterion under which instability occurs is presented. Analytical expressions for the neutrino oscillation frequency and the growth rate are obtained. The possible relevance of our investigation to non-thermal neutrino oscillations in dense plasma environments (e.g. the supernovae, the core of white dwarf stars etc.) is briefly mentioned.Neutrinos, which are the least massive subatomic ghostly elementary particles in the set of building blocks of nature, were produced in the Big-Bang, which began the universe, and are emitted by the Sun and all other stars (e.g. neutron stars, white dwarf stars, magnetars etc.) These are also produced in violent astrophysical events in cosmos, e.g. explosions of supernovae, which occur when an old massive star collapses after running out of nuclear fuel. During the collapse, the star literally becomes a neutron star as neutrinos totally dominate in numbers of particles for a few seconds and carry off most of the energy from the implosion, more energy than radiated during the entire life of the star. For example, the neutrino flash was actually observed in Supernova 1987A. Neutrinos are also created when cosmic rays, fast moving particles from space, bombard the Earth's atmosphere producing cascades of secondary particles, which rain down on us. There are three different kinds, or 'flavors', as they are called, of neutrinos: electron, muon and tau neutrinos. There are also three anti-neutrinos of the same flavors. The neutrinos get their names from their charged lepton brethren, the electron, muon and tau, with masses of 0.511 MeV, 106 MeV and 1777 Mev, respectively. The three species of neutrinos at https://www.cambridge.org/core/terms. https://doi