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Differential operators, Help


Hi group,

This should be relatively easy, but after several tries I have not been able
to do it.

I want define a differential operator in the following way. Let Dx and Dx2
denote the operators for first and second order differentiation with respect
to x. I want P to be an operator which depends on x,  Dx and Dx2.

For example, with

                  P = Dx2 + x Dx

I want P[f[z],z] = f''[z] + z f'[z].

This much I can do, but I cannot find a method which also gives the
following result,

P[ P[f[z],z], z]  -> (f''[z] + z f'[z])''+ z (f''[z] + z f'[z])'

Or more generally, if I have two such operators P and Q I want the correct
result from expressions such as

      P[ Q[f[x],x], x]

Any suggestion will be greatly appreciated.

Bill






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