By J. A. Richards

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**Sample text**

27b) demonstrates how fl can be determined computationally in a very convenient manner, once the discrete transition matrix for a Hill equation is known. Second order systems with losses lead to an equation with a first derivative. This is easily handled using the transformation of Eq. 4), and then adopting the techniques just outlined. Alternatively Eqs. 28) and fll + flz = -

24b) demonstrates that one of each type of P solution will exist for a lossless second order system, if the eigenvalues of the discrete transition matrix are positive. For an N type solution the characteristic exponent is seen from Eq. 22) to be of the form fl = rx + j(2n + 1)~, n-integer. Usually the choice n = 0 is made so that the associated basis solution is The Equations and Their Properties 24 e~tej"t/82(t). Again this is seen to be an exponentially modulated periodic function, with the direction of the modulation dependent upon the sign of rx.

5b) Here the significant time intervals have been seen to be (0, T) and (T, n). In a similar manner the transition matrix over an interval smaller than n can be determined. 3 Solution at Any Time for a Second Order Periodic Equation It is instructive to continue the above example. However the analysis to follow holds for any second-order (Hill) equation and thus it is useful to separate it. 7) to follow apply also to higher order equations. At a time t = mn + X, X < n, m-integral, the solution to a homogeneous periodic differential equation can be seen from Eqs.