Originally Posted by

**mezy** I am trying to figure out the concept behind inverse operators, but I am getting stuck. My book gives the definition as:

$\displaystyle y_{p} = \frac{b}{a_{0}}(1 + b_{1}D + b_{2}D^{2} + \ldots + b_{k}D^{k})x^{k}), where a_{0} \neq 0$

where $\displaystyle (1 + b_{1}D + b_{2}D^{2} + \ldots + b_{k}D^{k})x^{k})$ is the series expansion of the inverse operator 1/P(D) obtained by ordinary division.

Okay - that sounds great, but what are they dividing by?

Here is the example in the book:

$\displaystyle 4y'' - 3y' + 9y = 5x^{2}, P(D) = 4D^{2} - 3D + 9$

$\displaystyle Step 1: y_{p} = \frac{1}{9(1-\frac{D}{3}+\frac{4D^{2}}{9})}(5x^{2})$

$\displaystyle Step 2: = \frac{5}{9}(1+\frac{D}{3}-\frac{D^2}{3})x^{2}$

This is where I get lost. How did they go from step 1 to step 2?

Any help is greatly appreciated!