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my problem is messy, and I will use a simple problem to explain my problem. Suppose, we have a set of DE:

eq1=u''[x]+v[x]
eq2=v''[x]+u[x]
eq3=u[x]*u''[x]-v[x]*v''[x]

This is how I treat it in Mathematica. The actual set of DE consists of 3 equations eq1==0, eq2==0, eq3==0. It is easy to see, that we don't need 3rd equation here, since we can obtain it by multiplying first DE by u[x], second DE by v[x], and subtracting second DE from the first DE. I want to know, how to figure out using mathematica, if one of the equations is redundant.

My actual problem is more complicated. It has different parameters, and equations are non-linear (there are terms like Cos[v[x]]), coefficients are nonlinear as well.

Is there some "general" method to implement this procedure in Mathematica?

Thanks in advance.

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    $\begingroup$ Well, you need as many equations as functions and making sure that all functions are used in the subset. Of course not every subset will present the same difficulty ... $\endgroup$ Oct 8, 2015 at 16:13
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    $\begingroup$ I'm voting to close this question as off-topic because the issue it raises is not a Mathematica issue but a mathematical one. That it is formulated in terms of Mathematica is not sufficient to make it an appropriate question for Mathematica.SE. $\endgroup$
    – m_goldberg
    Oct 8, 2015 at 18:28
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    $\begingroup$ It is a reasonable question as to how to do this in Mathematica. I make that claim because it falls into the area of differential algebra, hence may be amenable to algorithmic approaches. $\endgroup$ Oct 8, 2015 at 22:02

1 Answer 1

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This might be along the lines of what you want.

eqns = {u''[x] + v[x], v''[x] + u[x], u[x]*u''[x] - v[x]*v''[x]};
vars = {u[x], v[x], u''[x], v''[x]};
GroebnerBasis[eqns, vars]

(* Out[52]= {v[x] + (u^\[Prime]\[Prime])[x], 
 u[x] + (v^\[Prime]\[Prime])[x]} *)

In general you may need to do more work to get enough equations, e.g. using prolongation.

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