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Inequalities for solutions of linear differential equations a contribution to the theory of servomechanisms
Published online by Cambridge University Press: 31 October 2008
Extract
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.
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- Copyright © Edinburgh Mathematical Society 1952
References
1 The author has previously shown that the result is true, more generally, if c 1 is a continuous non-vanishing function of x: see Zeitschrift für angewandte Mathematik und Mechanik, 38 (1942), 143–152 (Hilfssate, 144).Google Scholar
1 Only trivial modifications are required if the roots are all real or all unreal.
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