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A new linear integro-differential equation of order n ≥ 3, given on a closed curve located in the complex plane, is investigated. Integrals in the equation are understood in the sense of the finite part according to Hadamard. A characteristic feature of the equation is the presence of quadratic functions of a special kind in its coefficients. The equation is reduced first to the boundary value problem of linear conjugation for analytical functions. In the case of its solvability, two linear differential equations should be further solved in the domains of the complex plane with some additional conditions for the solution. All conditions for the solvability of the original equation are explicitly specified. When they are executed, the desired solution is constructed in a closed form. An example is given.
А. П. Шилин (Tue,) studied this question.