G. Kharatishvili

Institute of Cibernetics, Georgian Academy of Sciences Tbilisi, Georgia

FORMULAE FOR THE GENERAL SOLUTIONS FOR SOME CLASSES OF LINEAR NONHOMOGENEOUS

DIFFERENTIAL EQUATIONS OF SECOND ORDER WITH VARIABLE COEFFICIENTS

Let us consider the equation

α(t)x′′ + β(t)x′ + γ(t)x = δ(t), t_{1} < t < t_{2},  (1)

where α, β, γ, δ are given continuous functions.

The function is called possible if the above operations are correct.

Theorem 1. If the following conditions are satisfied

α(t) ≡ 1, β(t) ≡ 0, γ(t) = − \frac{η′′(t)}{η(t)} , t_{1} < t < t_{2},  (2)

where η is any possible function, then the general solution of the equation (1) has the following form

x = η(c_{1} + ∫ \frac{1}{η^{2}} (c_{2} + ∫ ηδ dt) dt), t_{1} < t < t_{2}.  (3)

Let

η = e^{∫ ξ(t) dt}, t_{1} < t < t_{2},  (4)

where ξ is any possible function, then from the Theorem 1 follows

Theorem 2. If the following conditions are satisfied α(t) ≡ 1, β(t) ≡ 0, γ(t) = −(ξ′(t)+ξ^{2}(t)), t_{1} < t < t_{2}, then the general solution of the equation (1) is given by (3), where the function η is determined by (4).

There also is much general result.

Theorem 3. If the following conditions are satisfied

\begin{cases}ab = α(t) \\ ab′ + qb + ah = β(t) \\ ah′ + qh = γ(t), t_{1} < t < t_{2} \end{cases} ,

where a = a(t), b = b(t), q = q(t), h = h(t) are any possible functions, then the general solution of the equation (1) is represented in finite terms by (3).

Finally, we note that Theorem 2 generalizes the results of M. Elshin (Bull. Acad. Sci. USSR, XVIII(1938)) and has important applications. For example if ξ(t) = Ae^{ωt} + B, where A, B, ω are constants, then from the equation (1), assuming α(t) ≡ 1 and β(t) ≡ 0, one gets generalized Hille’s type equation.