Using the D operator without true understanding.

[Matternot]

I was introduced to the d operator to help solve constant coefficient differential equations for the particular integral without using trial solutions:

http://en.wikipedia.org/wiki/Differential_operator
http://en.wikipedia.org/wiki/Shift_theorem

The results generally seem sensible, but there are a few results that aren't intuitive

e.g. (1-D)^-1 can be expanded to (1+D+D²+...) as though it were any number on (-1,1)

How can you show something like this to be mathematically viable.

I have so far been using results that haven't been proven to me. It feels like black magic!

Thanks

 

[robphy]

In my experience, it helps to first check things out by understanding what happens when
the operator is being applied to an arbirtrary function f.
Then, after seeing the result, then one can look at the pattern.
In your mind's eye, think of the function being there... but you probably don't need to write it out all the time.

 

[Matternot]

But I have no other way of performing (1-D)^-1 on a function and thus have no other way to compare to. I was always taught to just expand binomially and therefore do not know how to perform (1-D)^-1 otherwise.

 

[robphy]

This might help
http://math.stackexchange.com/questions/30020/inverse-operator-methods-for-differential-equations
http://www.efunda.com/math/ode/linearode_invoperator.cfm

 

[Matternot]

Thanks a lot! This looks great! I'll have a read through