Consider the nth order differential equation
where the coefficients cv are real constants and f is a real function continuous in the interval a≦ x ≦ b. The following theorem will be proved in §4:
If the characteristic equation of (I) has no purely imaginary roots, then a particular integral η (x) can always be found which satisfies the inequality
where C is a certain function of the cv only and M is the maximum of |f|. In particular we may take C = 1 if all roots of the characteristic equation are real.