A method for solving the Lagrange problem with state variable constraints for processes described by ordinary di erential equations without involvement of the Lagrange principle is proposed. Namely, dynamical system described by the equation.A necessary and sucient condition for existence of a solution to the variational problem is obtained, an admissible control is found and an optimal solution is constructed by narrowing a set of admissible controls. The basis of the proposed method for solving the variational problem is an imbedding principle. An essence of the imbedding principle is that the original variational problem with boundary conditions and state variable constraints is replaced by equivalent free end point optimal control problem. This approach is possible due to nding a general solution of a class of the rst kind Fredholm integral equations.