How I can use Groebner bases for reduction of differential polunomials?

Question

For Set of partial differential equations

${\frac {\partial }{\partial u}}\xi \left( x,t,u \right) =0$

${\frac {\partial }{\partial u}}\tau \left( x,t,u \right) =0$

$2\,{\frac {\partial }{\partial x}}\tau \left( x,t,u \right) -2\,{ \frac {\partial }{\partial t}}\xi \left( x,t,u \right) =0 $

$2\,{\frac {\partial }{\partial x}}\xi \left( x,t,u \right) -2\,{\frac {\partial }{\partial t}}\tau \left( x,t,u \right) =0 $

${\frac {\partial ^{2}}{\partial {u}^{2}}}\phi \left( x,t,u \right) =0$

${\frac {\partial ^{2}}{\partial {x}^{2}}}\xi \left( x,t,u \right) -2\, {\frac {\partial ^{2}}{\partial x\partial u}}\phi \left( x,t,u \right) -{\frac {\partial ^{2}}{\partial {t}^{2}}}\xi \left( x,t,u \right) =0 $

${\frac {\partial ^{2}}{\partial {x}^{2}}}\tau \left( x,t,u \right) +2 \,{\frac {\partial ^{2}}{\partial u\partial t}}\phi \left( x,t,u \right) -{\frac {\partial ^{2}}{\partial {t}^{2}}}\tau \left( x,t,u \right) =0 $

$-{\frac {\partial ^{2}}{\partial {x}^{2}}}\phi \left( x,t,u \right) +{ \frac {\partial ^{2}}{\partial {t}^{2}}}\phi \left( x,t,u \right) - \phi \left( x,t,u \right) {\frac {\rm d}{{\rm d}u}}f \left( u \right) + \left( {\frac {\partial }{\partial u}}\phi \left( x,t,u \right) \right) f \left( u \right) -2\, \left( {\frac {\partial }{\partial t} }\tau \left( x,t,u \right) \right) f \left( u \right) =0 $

By applying Groebner bases how I can derive constraint on $f(u)$ given as below

${\frac {{\rm d}^{3}}{{\rm d}{u}^{3}}}f \left( u \right) =-{\frac {-2\, \left( {\frac {{\rm d}^{2}}{{\rm d}{u}^{2}}}f \left( u \right) \right) ^{2}f \left( u \right) + \left( {\frac {{\rm d}^{2}}{{\rm d}{ u}^{2}}}f \left( u \right) \right) \left( {\frac {\rm d}{{\rm d}u}}f \left( u \right) \right) ^{2}}{ \left( {\frac {\rm d}{{\rm d}u}}f \left( u \right) \right) f \left( u \right) }} $

Answer 1

This can be easily with the aid of Maple package "DifferentialAlgebra". Use following commands:

with(DifferentialAlgebra)

R := DifferentialRing(blocks = [xi, tau, phi, f], derivations = [x, t, u])

G := RosenfeldGroebner(DetSys, R)

Equations(G[2], solved)

The last command will give you constraint equation on $f(u)$.