ABSTRACT

In the case of a linear equation with constant coefficients all particular integrals are elementary functions, both for the unforced equation, and for the forced equation with forcing by an elementary function; the complete integral can be derived from the roots of characteristic polynomial. The characteristic polynomial can be used to obtain solutions of the homogeneous linear differential equation with power coefficients, both unforced and forced. A linear unforced differential equation with constant coefficient consists of a characteristic polynomial of derivatives applied to the dependent variable. An alternative to the use of the characteristic polynomial is to substitute in the differential equation a trial solution with some coefficients to be determined. A power or polynomial forcing function does not lead to a resonant particular integral unless the characteristic polynomial has a root zero. The method of the inverse characteristic polynomial applies in the cases I(II) of linear differential equations with constant coefficients to forcing by an arbitrary smooth function.