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Consequently the DE 3 is actually the following system of ODE in the functions t...

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Algebra

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Consequently the DE 3 is actually the following system of ODE in the functions t X t t X t t X t d dX dk X 1 X 5 X dik dt dk X dik 12 1 X 4 dk X di di 1 d x 1 1 x dx d dX dik 1 X 1 dX dx dt d X dik t At this stage we become more specific about the features of the ODE 3 or equivalently about the system 4 Let I be an open internal and let denote an open subset of R We consider the open sets IxNxR x xR there being k 1 copies of R in the above Cartesian product This set is being designated to accommodate the variable quantities dk X di 1 dX dt 4 Clearly the function f appearing on the right hand side of 3 must have this set as its domain of definition Choosing 1 1 x En and w w win R we form the initial condition fo W W w Now the initial value problem for the ODE 3 is the following pair dk X di 1 ff x ToX W W 1 5 By a solution of the initial value problem 5 we mean an at least k mously diffamanti 01

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