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Bucking of a thin flexible elongated plate in supersonic gas flow along Ox-axis, compressed or extended by boundary stresses at the edges x = 0 and x = 1 is governed by a boundary value problem for nonlinear ordinary integro-differential equation with two bifurcation parameters: external stress and Mach number. The linearized equation for the described boundary problem of aeroelasticity [2] χ 2 w (4)= 0 with boundary conditions B or B representing spectral two-point boundary value problem. Here χ 2 rigidity of the plate, T < 0 (T > 0), proportional to compressing stress (extending stress), σ > 0, proportional to Mach number are dimensionless bifurcation parameters. At the investigation of these boundary value problems the following possibilities arise:In the first case T is greater then 0. The characteristic equation (Ch.Eq.) f 0 (λ) = χ 2 λ 4 − T λ 2 + σλ = 0 has one negative root −α and two positive roots β 2 > β 1 > 0 (α = β 1 + β 2 ). For d = 0 T is greater than 0 again. Here there are two equal roots β 1 = β 2 = β > 0 and one negative root −α. Third case which is possible at both extension (T > 0) and compression (T < 0) of the plate, the roots are a pair of complex conjugate γ ± δi (γ, δ > 0) and −α < 0 (α=2γ). Here for the buckling investigation it is convenient to introduce the following. It is not difficult to see that the values3 χ 2 respond to the plate extension, and the values u > √ 3 ⇒ σ > 8γ 3 χ 2 respond to the plate compression. The value u= √ 3 implies T =0, i.e. the extension/compression absence. The value u = 0 corresponds to d = 0. At the investigation of Ch.Eq. with two parameters T and σ Sturm method for roots separation was used.

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At the application of bifurcation theory methods to nonlinear boundary value problems for ordinary dierential equations of the fourth and higher order there usually arise technical diculties, connected with determination of bifurcation manifolds, spectral investigation of the direct and conjugate linearized problems and the proof of their Fredholm property. For overcoming of this diculty here the roots separation method is applied to the relevant characteristic equations with subsequent presentation of critical manifolds, that allows to investigate nonlinear problems in the precise statement. Such approach is applied here to two-point boundary value problem for the nonlinear ODE of the fourth order describing the buckling (divergence) of an elongated plate in a supersonic ow of gas, subjected to compressed or extended boundary stresses at the various boundary fastenings.