On multidimensional determinant differential-operator equations

We consider a class of multi-dimensional determinant differential-operator equations, the left side of which represents a determinant with the elements containing a product of linear one-dimensional differential operators of arbitrary order, while the right side of the equation depends on the unknown function and its first derivatives. The homogeneous and inhomogeneous determinant differential-operator equations are investigated separately. Some theorems on decreasing of dimension of equation are proved. The solutions obtained in the form of~sum and product of functions in subsets of independent variables, in particular, of~functions in one variable. In particular, it is proved that the solution of the equation under considering is the product of eigenfunctions of linear operators contained in the equation. A theorem on interconnection between the solutions of the initial equation and the solutions of some auxiliary linear equation is proved for the homogeneous equation...