ABSTRACT
“Variation of parameters”, which is basically a clever refinement of the reduction of order method, is another way to solve nonhomogeneous linear differential equations. Unlike the method of undetermined coefficients, the coefficients of the differential equation do not need to be constants, nor does the forcing function need to be of a particular type. The coefficients and forcing function need only be “sufficiently smooth” on the interval of interest.
This chapter begins with the derivation via reduction of order of the variation of parameters method for second-order nonhomogeneous linear differential equations. It is shown that the method can be reduced, essentially, to the integration of the solutions to a relatively simple system of two linear equations. The extension of this method to handle higher-order nonhomogeneous equations then follows in a very natural manner. The chapter finishes by noting that the variation of parameters method can be reduced to a single integral formula involving Wronskians. The advantages and disadvantages of this formula, as well as the advantages and disadvantages of using the definite or the indefinite integral is also discussed.
