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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![rubel := 3*y[x]^4*y[x, x]*y[x, x, x, x]^2-4*y[x]^4*y[x, x, x]^2*y[x, x, x, x]+6*y[x]^3*y[x, x]^2*y[x, x, x]*y[x, x, x, x]+24*y[x]^2*y[x, x]^4*y[x, x, x, x]-12*y[x]^3*y[x, x]*y[x, x, x]^3-29*y[x]^2*y[x, x]^3*y[x, x, x]^2+12*y[x, x]^7](/support/helpjp/helpview.aspx?si=7388/file07259/math393.png)
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![[[3*y[x]^4*y[x, x]*y[x, x, x, x]^2-4*y[x]^4*y[x, x, x]^2*y[x, x, x, x]+6*y[x]^3*y[x, x]^2*y[x, x, x]*y[x, x, x, x]+24*y[x]^2*y[x, x]^4*y[x, x, x, x]-12*y[x]^3*y[x, x]*y[x, x, x]^3-29*y[x]^2*y[x, x]^3*y[x, x, x]^2+12*y[x, x]^7], [y[x, x, x]^2*y[x]^2+3*y[x, x]^4], [y[x, x]]]](/support/helpjp/helpview.aspx?si=7388/file07259/math400.png)
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![[[3*y[x]^4*y[x, x]*y[x, x, x, x]^2-4*y[x]^4*y[x, x, x]^2*y[x, x, x, x]+6*y[x]^3*y[x, x]^2*y[x, x, x]*y[x, x, x, x]+24*y[x]^2*y[x, x]^4*y[x, x, x, x]-12*y[x]^3*y[x, x]*y[x, x, x]^3-29*y[x]^2*y[x, x]^3*y[x, x, x]^2+12*y[x, x]^7], [y[x, x, x]^2*y[x]^2+3*y[x, x]^4]]](/support/helpjp/helpview.aspx?si=7388/file07259/math407.png)
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