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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![R := differential_ring(derivations = [t], ranking = [y], notation = diff)](/support/helpjp/helpview.aspx?si=7388/file07259/math117.png)
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| (1) |
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![equations(Rosenfeld_Groebner([p], R))](/support/helpjp/helpview.aspx?si=7388/file07259/math128.png)
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![[[(diff(y(t), t))^3-4*t*y(t)*(diff(y(t), t))+8*y(t)^2], [27*y(t)-4*t^3], [y(t)]]](/support/helpjp/helpview.aspx?si=7388/file07259/math131.png)
| (2) |
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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![[[(diff(y(t), t))^3-4*t*y(t)*(diff(y(t), t))+8*y(t)^2], [27*y(t)-4*t^3]]](/support/helpjp/helpview.aspx?si=7388/file07259/math154.png)
| (3) |
Let us consider the two similar differential polynomials 
and 
.
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![R := differential_ring(derivations = [t], ranking = [y])](/support/helpjp/helpview.aspx?si=7388/file07259/math164.png)
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![p := y[t]^2-4*y[]](/support/helpjp/helpview.aspx?si=7388/file07259/math168.png)
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![p := y[t]^2-4*y[]](/support/helpjp/helpview.aspx?si=7388/file07259/math171.png)
| (4) |
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![q := y[t]^2-4*y[]^3](/support/helpjp/helpview.aspx?si=7388/file07259/math175.png)
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![q := y[t]^2-4*y[]^3](/support/helpjp/helpview.aspx?si=7388/file07259/math178.png)
| (5) |
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![Cp := equations(Rosenfeld_Groebner([p], R))](/support/helpjp/helpview.aspx?si=7388/file07259/math182.png)
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![Cp := [[y[t]^2-4*y[]], [y[]]]](/support/helpjp/helpview.aspx?si=7388/file07259/math185.png)
| (6) |
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![Cq := equations(Rosenfeld_Groebner([q], R))](/support/helpjp/helpview.aspx?si=7388/file07259/math189.png)
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![Cq := [[y[t]^2-4*y[]^3], [y[]]]](/support/helpjp/helpview.aspx?si=7388/file07259/math192.png)
| (7) |
Both 
and 
admit 
as a singular zero. Nonetheless:
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![Mp := [[y[t]^2-4*y[]], [y[]]]](/support/helpjp/helpview.aspx?si=7388/file07259/math207.png)
| (8) |
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![Mq := [[y[t]^2-4*y[]^3]]](/support/helpjp/helpview.aspx?si=7388/file07259/math214.png)
| (9) |
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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![R := differential_ring(derivations = [x, y], ranking = [u])](/support/helpjp/helpview.aspx?si=7388/file07259/math245.png)
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![p := -u[]+y*u[y]+x*u[x]-u[x]^2-u[y]^2](/support/helpjp/helpview.aspx?si=7388/file07259/math249.png)
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![p := -u[]+y*u[y]+x*u[x]-u[x]^2-u[y]^2](/support/helpjp/helpview.aspx?si=7388/file07259/math252.png)
| (10) |
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![C := equations(Rosenfeld_Groebner([p], R))](/support/helpjp/helpview.aspx?si=7388/file07259/math256.png)
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![C := [[u[x]^2-x*u[x]+u[y]^2-y*u[y]+u[]], [2*u[x]-x, 4*u[y]^2-4*y*u[y]+4*u[]-x^2], [4*u[]-x^2-y^2]]](/support/helpjp/helpview.aspx?si=7388/file07259/math259.png)
| (11) |
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![M := [[u[x]^2-x*u[x]+u[y]^2-y*u[y]+u[]], [4*u[]-x^2-y^2]]](/support/helpjp/helpview.aspx?si=7388/file07259/math266.png)
| (12) |
A differential polynomial in several variables:
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![R := differential_ring(derivations = [x, y], ranking = [u, v])](/support/helpjp/helpview.aspx?si=7388/file07259/math276.png)
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![p := u[x, y]^2*v[y]-u[x, y]*v[y]*u[y]-u[y]*u[x, y]+u[y]^2](/support/helpjp/helpview.aspx?si=7388/file07259/math280.png)
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![p := u[x, y]^2*v[y]-u[x, y]*v[y]*u[y]-u[y]*u[x, y]+u[y]^2](/support/helpjp/helpview.aspx?si=7388/file07259/math283.png)
| (13) |
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![Rosenfeld_Groebner([p], R)](/support/helpjp/helpview.aspx?si=7388/file07259/math287.png)
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![[characterizable, characterizable]](/support/helpjp/helpview.aspx?si=7388/file07259/math290.png)
| (14) |
It would seem that there several types of zeros, the general zero of 
and several singular zeros. Nonetheless,
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![MR := [characterizable]](/support/helpjp/helpview.aspx?si=7388/file07259/math301.png)
| (15) |
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![ER := [[u[x, y]^2*v[y]-u[x, y]*v[y]*u[y]-u[y]*u[x, y]+u[y]^2]]](/support/helpjp/helpview.aspx?si=7388/file07259/math308.png)
| (16) |
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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![Q := differential_ring(derivations = [x, y], ranking = [v, u])](/support/helpjp/helpview.aspx?si=7388/file07259/math320.png)
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![MQ := [characterizable]](/support/helpjp/helpview.aspx?si=7388/file07259/math327.png)
| (17) |
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![EQ := [[v[y]*u[x, y]-u[y]]]](/support/helpjp/helpview.aspx?si=7388/file07259/math334.png)
| (18) |
We check that the two differential polynomials appearing in this decompositions are the two factors of differential polynomials appearing in 
.
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![factor(ER[1][1])](/support/helpjp/helpview.aspx?si=7388/file07259/math342.png)
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![(-u[x, y]+u[y])*(-v[y]*u[x, y]+u[y])](/support/helpjp/helpview.aspx?si=7388/file07259/math345.png)
| (19) |
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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![R := differential_ring(ranking = [y], derivations = [x])](/support/helpjp/helpview.aspx?si=7388/file07259/math353.png)
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![chazy := -(y[x, x]+y[]^3*y[x])^2+(y[]*y[x])^2*(4*y[x]+y[]^4)](/support/helpjp/helpview.aspx?si=7388/file07259/math357.png)
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![chazy := -(y[x, x]+y[]^3*y[x])^2+y[]^2*y[x]^2*(4*y[x]+y[]^4)](/support/helpjp/helpview.aspx?si=7388/file07259/math360.png)
| (20) |
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![equations(Rosenfeld_Groebner([chazy], R))](/support/helpjp/helpview.aspx?si=7388/file07259/math364.png)
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![[[y[x, x]^2+2*y[x, x]*y[]^3*y[x]-4*y[]^2*y[x]^3], [4*y[x]+y[]^4], [y[x]]]](/support/helpjp/helpview.aspx?si=7388/file07259/math367.png)
| (21) |
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![[[y[x, x]^2+2*y[x, x]*y[]^3*y[x]-4*y[]^2*y[x]^3], [4*y[x]+y[]^4]]](/support/helpjp/helpview.aspx?si=7388/file07259/math374.png)
| (22) |
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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![R := differential_ring(ranking = [y, z], derivations = [x])](/support/helpjp/helpview.aspx?si=7388/file07259/math386.png)
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![rubel := 3*y[x]^4*y[`$`(x, 2)]*y[`$`(x, 4)]^2-4*y[x]^4*y[`$`(x, 3)]^2*y[`$`(x, 4)]+6*y[x]^3*y[`$`(x, 2)]^2*y[`$`(x, 3)]*y[`$`(x, 4)]+24*y[x]^2*y[`$`(x, 2)]^4*y[`$`(x, 4)]-12*y[x]^3*y[`$`(x, 2)]*y[`$`(x, 3)]^3-29*y[x]^2*y[`$`(x, 2)]^3*y[`$`(x, 3)]^2+12*y[`$`(x, 2)]^7](/support/helpjp/helpview.aspx?si=7388/file07259/math390.png)
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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)
| (23) |
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![equations(Rosenfeld_Groebner([rubel], R))](/support/helpjp/helpview.aspx?si=7388/file07259/math397.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)
| (24) |
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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)
| (25) |
