The boundary value problem for second order ordinary linear differential equations with variable coefficients

In this work two-point boundary value problem for one class of second order ordinary differential equations with variable coefficients is solved.


Introduction
We consider the equation Two-point boundary value problems for second order ordinary differential equations are classical area of research of the theory of ordinary differential equations and because of their broad application in mechanics, mathematical physics and geometry (see, for example,[1]- [9]) they are still actively investigated. However, in mathematical literature the equations of the form (1) with continuous coefficients are studied and sufficient conditions of resolvability of boundary value problems for them are received. In author's works [10], [11] the general solution of equation (1) is constructed and Cauchy problem for it with initial point x = 0 is solved. In this work an explicit form of general solution of equation (1) in class where is found and next boundary value problem is solved. Problem D. Find the solution of equation (1) from the class (2) satisfying the conditions where α, β are given real numbers.
2 Construction of the general solutions to equation (1) By integrating two times the equation (1), we get where c 1 , c 2 are any real numbers, Applying the operator B to both sides of equation (5) we have where From (5) and (6) it follows Further we use following functions and operators: Applying the operator B to both sides of equation (7) we get From (5) and (8) it follows Continuing this procedure n times we obtain the following integral representation of solutions of equation (1): where (B 0 g)(x) = g(x). Let u(x) ∈ C[0, x 1 ]. The following inequalities are easily obtained: where |f | 1 = max x∈[0, x 1 ] |f (x)|.
Passing to the limit with n → ∞ in the representation (9) and taking inequalities (10), (3) into account we get where Using the inequalities (10), (11), we receive From the form of functions I 1 (x), Considering (14) and the form of functions I 1 (x), I 2 (x), F (x) we get From (15) and (16) it follows, that functions I 1 (x), I 2 (x) are particular solutions from class (2) of homogeneous equations d 2 u dx 2 + a(x)u = 0, and the function F (x) is solution of non-homogeneous equation (1).
From (15) and (17) we see that the Wronskian W (x) is equal to −I 2 (x 1 ). Therefore if I 2 (x 1 ) = 0 then the functions I 1 (x) and I 2 (x) are linear independent on [0, x 1 ] and the general solution to equation (1) is determined by the formula (12).
Hence, we proved the following theorem.

Solution of boundary value problem
For equation (1) we consider the problem D. To solve the problem D we use the solution of equation (1), given by the formula (12). Substituting the function u(x), given by formula (12), into boundary conditions (4) and taking into account (17) we have Hence, the solution of problem D has a form u(x) = βI 1 (x) + αI 2 (x) + F (x).
Therefore, the following theorem is proved.