Inequalities for solutions of linear differential equations a contribution to the theory of servomechanisms

Consider the n th order differential equation where the coefficients c v 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 c v only and M is the maximum of |f|. In particular we may take C = 1 if all roots of the characteristic equation are real.