Consider a linear non-homogeneous ordinary differential equation with constant coefficients

where are all constants and . Let

the ODE can be rewritten as

Since all coefficients are constants, the above equation can be factored into

Thus,

The particular solution can be obtained by repeated integration of these inverse differential operators.

Pros and Cons of the Method of Reduction of Order: The method of reduction of order is very straightforward but not always easy to perform unless all are real numbers. In addition, n integrations in sequence are not convenient to check.

Modification of the Method of Reduction of Order: By performing the partial fraction expansion, the sequential integration can be broken into the sum of a serial individual integrations, i.e.,

If are k repeated roots, the particular solution becomes