In previous papers, a generalization of the well-known Euler equation with an arbitrary differentiable generating function was proposed. Criteria are formulated that allow direct integration of inhomogeneous equations, bypassing the well-known Lagrange method of variation of arbitrary constants. The disadvantage of the method is the necessity of $n$-fold integration of the equation. The present paper considers the idea of replacing the $n$-fold integration with a system of linear equations obtained by a single integration of the original $n$-order equation with the found roots of the characteristic equation.