I know how the process works, first from difference equation make a homogeneous equation and then find natural response , then find a particular response (in the form of input) substitute it back in the equation and find constants. And we get a forced response. I have a doubt in finding constants.
This example is Drill problem 2.12 in "Signals and Systems" by Simon Haykin:
For the equation: $$y[n] - (0.25)*y[n-2] = 2*x[n]+x[n-1]$$
find a forced response for $x[n]=u[n].$
I have found the natural response and for particular response. Say, $y[n] = k*u[n]$
Now the equation is:
$$k*u[n] - 0.25*k*u[n-2] = 2*u[n] + u[n-1]$$
The answer according to the book is $k=4$, but that is true only for $n>=2$. Can somebody explain what happens if $2>n>0$; $k=4$ does not seem correct in this case, as $u[n-2]=0$ for $n<2$ and $u[n-1]=0$ for $n<1$.
Why can we just substitute all $u[n-c] = 1$, in the difference equation for finding value of constants in particular solution?