- #1
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Hopefully the symbols I am using are standard. I will define them upon request.
I have a theorem that says, given a difference equation \(\displaystyle \left ( \sum_{j = 0}^m a_j E^j \right ) y_n = \alpha ^n F(n)\), we can define a polynomial function \(\displaystyle \phi (E) = \sum_{j = 0}^m a_j E^j \) such that \(\displaystyle \phi (E) y_n = \alpha ^n F(n)\). I can follow a proof to the following result:
(1) \(\displaystyle \phi (E) \left ( \alpha ^n F(n) \right ) = \alpha ^n \phi ( \alpha E ) F(n)\)
The notes then go on to say, "therefore"
(2) \(\displaystyle \dfrac{1}{ \phi (E) } \left ( \alpha ^n F(n) \right ) = \alpha ^n \dfrac{1}{ \phi ( \alpha E )} F(n)\)
Now, the particular solution to the difference equation is written as \(\displaystyle y_p = \dfrac{1}{ \phi (E) } \left ( \alpha ^n F(n) \right )\) and I can use (2) to evaluate this and get the correct result. So I know that (2) is right, without any typos. But how do I get from (1) to (2)?
More details upon request.
-Dan
I have a theorem that says, given a difference equation \(\displaystyle \left ( \sum_{j = 0}^m a_j E^j \right ) y_n = \alpha ^n F(n)\), we can define a polynomial function \(\displaystyle \phi (E) = \sum_{j = 0}^m a_j E^j \) such that \(\displaystyle \phi (E) y_n = \alpha ^n F(n)\). I can follow a proof to the following result:
(1) \(\displaystyle \phi (E) \left ( \alpha ^n F(n) \right ) = \alpha ^n \phi ( \alpha E ) F(n)\)
The notes then go on to say, "therefore"
(2) \(\displaystyle \dfrac{1}{ \phi (E) } \left ( \alpha ^n F(n) \right ) = \alpha ^n \dfrac{1}{ \phi ( \alpha E )} F(n)\)
Now, the particular solution to the difference equation is written as \(\displaystyle y_p = \dfrac{1}{ \phi (E) } \left ( \alpha ^n F(n) \right )\) and I can use (2) to evaluate this and get the correct result. So I know that (2) is right, without any typos. But how do I get from (1) to (2)?
More details upon request.
-Dan