G. Kharatishvili

Institute of Cybernetics, Georgian Academy of Sciences Tbilisi, Georgia

GENERAL SOLUTIONS FOR SOME CLASSES OF LAME GENERALIZED AND CLASSICAL EQUATIONS

Theorem 1. Let us consider the Lame generalized equation

α(t)ẍ + (αt^{2} + βt + γ)ẋ + (δt + η)x = f(t), α, β, γ, δ, η – ∀ const, (1)

where particular class is defined by

α = −δ, β = \frac{γδ}{η} − η.  (2)

General solution of equation (1) with condition (2) has the form

\begin{matrix}x = (t − \frac{γ}{η} )(c_{1} − ∫ (t − \frac{γ}{η} )^{−2}e^{− ∫ \frac{αt^{2}+βt+γ}{α(t)} dt}× \\ × (c_{2} − ∫ e^{∫ \frac{αt^{2}+βt+γ}{α(t)} dt}(t − \frac{γ}{η} )\frac{f(t)}{α(t)} dt)dt). \end{matrix}  (3)

If the following condition is satisfied

α(t) = (t − τ)(t − θ)(t − ξ), τ, θ, ξ – ∀ const,  (4)

then equation (1) pass into the Lame nonhomogeneous equation (algebraic form).

From Theorem 1 follows

Theorem 2. If condition (2) is satisfied, then the general solution of the Lame nonhomogeneous equation

(t − τ)(t − θ)(t − ξ)ẍ + (αt^{2} + βt + γ)ẋ + (δt + η)x = f(t)  (5)

has form (3), where α(t) has form (4).

Since

\frac{αt^{2} + βt + γ}{(t − τ)(t − θ)(t − ξ)} = \frac{k}{t − τ} + \frac{l}{t − θ} + \frac{m}{t − ξ} ,

where

k = \frac{ατ^{2} + βτ + γ}{(τ − θ)(τ − ξ)}, l = \frac{αθ^{2} + βθ + γ}{(θ − τ)(θ − ξ)}, m = \frac{αξ^{2} + βξ + γ}{(ξ − τ)(ξ − θ)},

formulae (3) has the form

\begin{matrix}x = (t − \frac{γ}{η} )(c_{1} − ∫ (t − τ)^{−k}(t − θ)^{−l}(t − ξ)^{−m}(t − \frac{γ}{η} )^{−2}× \\ × (c_{2} − ∫ (t − τ)^{k−1}(t − θ)^{l−1}(t − ξ)^{m−1}(t − \frac{γ}{η} )f(t) dt) dt). \end{matrix} (6)