Homogeneous differential-operator equations in locally convex spaces

We describe a general method that allows to find solutions to homogeneous differential-operator equations with variable coefficients by means of continuous vector-valued functions. The “homogeneity” is interpreted not in terms of null right-hand side of an equation but in terms that the left-hand side is homogeneous function of operators appearing in an equation. Solutions are presented by a uniformly convergent functional vector-valued series, generated by a set of solutions to some ordinary differential equation of $k$th degree, roots of characteristical polynomial, and some set of elements of locally convex space. We find sufficient conditions of continuous dependence of solution on generating set, and a solution to Cauchy's problem for considered equations. We specify conditions of its existence and uniqueness. Besides, under certain conditions we find a general solution to considered equation (a function of most general form from which any particular solution can be found). The investigation is realized by means of characteristics (order and type) of operator and operator characteristics (operator order and operator type) of vector relative to an operator. Also we use a convergence of operator series relative to equicontinuous bornology.