On the derivation of the formula for the general solution of an ordinary linear differential equation with constant coefficients

Background. Linear differential equations with constant coefficients are often used to solve a number of physical, engineering, chemical and biological problems. They are also used to solve and study more complex differential equations and, naturally, a significant number of scientific papers are devoted to them. The purpose of this note is to derive a formula for the general solution of a linear ordinary nonhomogeneous differential equation of the n-th order with constant coefficients in the form of one n-fold quadrature with respect to x, containing n arbitrary constants. Materials and methods. By multiplying a given linear ordinary nonhomogeneous differential equation of the n-th order with constant coefficients by a special function depending on x and then integrating, a new linear ordinary non-homogeneous differential equation with constant coefficients is obtained, equivalent to the original one and having an order less than n. Results. By applying the specified method repeatedly, but not more than n times, a formula for the general solution of the original linear ordinary nonhomogeneous differential equation of the n-th order with constant coefficients is derived in the form of one n-fold quadrature with respect to x, containing n arbitrary constants. A formula for a particular solution satisfying special conditions, generalizing the Cauchy formula, is also derived. Conclusions. The presented formula for the general solution of a linear ordinary non-homogeneous differential equation of the n-th order with constant coefficients in the form of one n-fold quadrature with respect to x, containing n arbitrary constants, complements the results devoted to the construction of a general solution of a linear ordinary differential equation of the n-th order with constant coefficients. In addition, the obtained formula for a particular solution satisfying special conditions is a generalization of the Cauchy formula.