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The first example illustrates how the Rosenfeld_Groebner command splits a system of differential equations into a system representing the general solution and systems representing the singular solutions.
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To obtain the characterizable differential ideal representing the general solution alone, we can proceed as follows.
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It is sometimes the case that the radical differential ideal generated by S is prime. This can be proved by exhibiting a ranking for which the characteristic decomposition of P consists of only one orthonomic characterizable differential ideal.
Before computing a representation of P with respect to the ranking of R, it may be useful to proceed as follows. Search for a ranking for which the characteristic decomposition is as described above. Assign J this computed characteristic decomposition. Then call Rosenfeld_Groebner with J as fourth parameter.
With such a fourth parameter, whatever the ranking of R is, the computed representation of P consists of only one characterizable differential ideal.
If J consists of a single nonorthonomic component or has more than one characterizable component, Rosenfeld_Groebner uses the information to avoid unnecessary splittings.
The example below illustrates this behavior for Euler's equations for an incompressible fluid in two dimensions.
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