Important: The diffalg package has been deprecated. Use the superseding package DifferentialAlgebra instead.
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Ordinary differential polynomials of first order:
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This differential polynomial has two singular zeros: the cubic and . Nonetheless, the general zero can be expressed as . Therefore, is a particular case () of the general solution. This is uncovered by essential_components without solving the differential equation. The function essential_components gives a minimal description of the zero set.
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Let us consider the two similar differential polynomials and .
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Both and admit as a singular zero. Nonetheless:
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is an essential singular zero of but not of . This has an analytic interpretation: is an envelope of the non singular zeros of while it is a limit of the non singular zeros of .
Incidentally: the general zero of can be expressed as . Thus, is a particular case of the general zero of .
Partial differential polynomials:
This illustrates the fact that the characteristic sets of the components of the minimal characteristic decomposition have only one element.
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A differential polynomial in several variables:
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It would seem that there several types of zeros, the general zero of and several singular zeros. Nonetheless,
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This show that the singular zeros exhibited by the Rosenfeld_Groebner decomposition are in fact particular zeros of the general zero of .
We illustrate now the fact that the underlying prime minimal decomposition of the obtained characteristic minimal decomposition is independent of the ranking.
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We check that the two differential polynomials appearing in this decompositions are the two factors of differential polynomials appearing in .
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Higher order differential polynomials:
The following equation arose in Chazy's work to extend the Painleve analysis to third order differential equations. In the process, he uncovered certain differential equations whose non-singular solutions have no movable singularity whereas one of the singular solutions does.
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The singular zeros are given by and . Only the second kind is essential.
The zeros of the following 4th order, homogeneous differential equation of degree 7 have the property that they can be used to approximate piecewisely any smooth function. This was shown by Rubel (1981).
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| (25) |