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 nonsingular 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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