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Explicit expressions for solutions of boundary-value problems and Cauchy problems related to the operator differential equation Xcn) + A,_ ,X("-') + . . . + A,X = 0 are given in terns of solutions of the algebraic operator equation X" + A, ,X"-' + . . . + A, = 0. A method for solving this algebraic equation is studied. For the scalar case, the equation (1.2) is always solvable, but this does not occur for the operator case. For instance, if A, is an operator without square *This work has beendoi:10.1016/0024-3795(88)90319-9 fatcat:reg5aqttefdjhpdp7dkdoycpgy