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Question: Linear PDE system, help!

Question:Linear PDE system, help!

alphard 5 Maple
linear solve
September 19 2010
1

Hi! I have a problem with the total PDE linear system:

F=F(x,y,z,w,h);

X1(F)=F[x]+F[y]+F[z]+F[w]=0;

X2(F)=x*F[x]+y*F[y]+z*F[z]+w*F[w] - h*F[h]=0;

X3(F)=x^2*F[x]+y^2*F[y]+z^2*F[z]+w^2*F[w] - ((z-x)*h+C*(w-y)/(z-x))F[h]=0;  (C=const)

The Poisson brackets are:

[X1,X2]=X1, [X1,X3]=2*X2, [X2,X3]=X3.

I'v tried to solve this system with pdsolve, but F=const is the only Maple's answer. But I am sure that this system has a more complicated solution. According to the X-operators' algebra, there must be solution for X1,X3 eqations at least..

Could someone help me to solve this system with maple or without it? Useful advises will be be good too..
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