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Linear Operators and the General Solution of Elementary Linear Ordinary Differential Equations

We make use of linear operators to derive the formulae for the general solution of elementary linear scalar ordinary differential equations of order n. The key lies in the factorization of the linear operators in terms of first-order operators. These first-order operators are then integrated by applying their corresponding integral operators. This leads to the solution formulae for both homogeneous- and nonhomogeneous linear differential equations in a natural way without the need for any ansatz (or "educated guess"). For second-order linear equations with nonconstant coefficients, the condition of the factorization is given in terms of Riccati equations.