In the first part of the present paper we deal with the first boundary value problem for general second-order differential equation in plane angle. The criterion of non-trivial solvability is obtained for such problem in space C2 of functions having polynomial growth at infinity. In the second part so-called "almost Cauchy" problem in a polygon for high order differential equation without respect of type is investigated. The necessary condition of uniqueness violation of solution is appeared to be sufficient in case of problem with one boundary condition.