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11) which is the required general solution. It is seen in the above that the Wronskian must be nonsingular, the condition noted earlier for the linear independence of the Xi(t). 3 The State Transition Matrix In Eq. 11) it is usual to write o/(t,O) = Wet) W-l(O). 12) The Equations and Their Properties 20 pet, 0) is referred to as thelstate transition matrix lover the interval (0, t). 11)canberewrittenas x(t) = ¢(t, O)x(O) + t ¢(t, r)f(r)dr. 13) Clearly once the transition matrices have been determined the complete solution to Eq.

2 for one interval of the rectangular waveform in the Meissner equation. The success of the Laplace transform in this derivation lies in the fact that the exponential kernel of the transformation is matched by the 'exponential' basis functions of the constant coefficient system. It is possible that other transformations could be used to deduce state transition matrices over intervals in which coefficients are time varying, provided again the kernels of those transforms match the basis function appropriate to the equation with that time variation.

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