Why "run the filter longer than needed and remove the initial values" will solve the issue of recursive solving equations? Consider sequence of random variables $w_i$ iid normal(0,1). Given the equation, $x_t=x_{t-1}-0.9x_{t-2}+w_t$ with $t$ discrete, I want to solve for $x_t$ recursively by prescribing $x_1,x_2$. The filter function is the filter function in $R$. 
"The data are obtained by a filter of white noise. The function filter uses zeros for the initial values. In this case, $x_1 = w_1$, and $x_2 = x_1 + w_2 = w_1 + w_2$, and so on, so that the values do not satisfy (1.2). An easy fix is to run the filter for longer than needed and remove the initial values."
Below is $R$ code:
w = rnorm(550,0,1) # 50 extra to avoid startup problems
x = filter(w, filter=c(1,-.9), method="recursive")[-(1:50)] # remove first 50
$\textbf{Q:}$ Since filter sets initial condition to $0$, it is clear that $x_1=w_1,x_2=w_1+w_2$.(i.e. Set $x_0=0,x_{-1}=0$.) However, knowing $x_1,x_2$, filter function with given appropriate coefficient, should interpret $x_3=x_2-0.9x_1+w_3$. Why " An easy fix is to run the filter for longer than needed and remove the initial values"?
 A: Hi: When you call the filter function in R, you're simulating the AR(2) model described at the top of your question. This statement of course assumes that you use the right function arguments so that you get the AR(2). There are parameters such as method, sides,  circular, init etc and these need to be set correctly in order to simulate the AR(2).
All the statement about running it long enough means is that, if you run it with a large enough sample, then the initial values won't have effect on the later values in the constructed series. Since the initial values don't satisfy the AR(2), you don't want them to have any effect on other values or even be contained in the simulated series.
So, for example, you can call filter(...) and run it for say 500 observations. Then cut off the first 50 and keep only the last 450. This way, whatever it was initialized with, has no effect and you'll have an "uncorrupted" AR(2). But, as I said, be careful when calling R's filter function.  It will give you results and it's upto you to check whether it's doing what you want it to be doing. I made mistakes calling it once and didn't realize that I was calling it incorrectly until way later and it cost me. Good luck.
