Difference equation

finding this pretty difficult

i've learnt difference equation solved using the shift operator E

so i began by rewriting

$y_{t} = 1.05y_{t-1} - D$

as
$(E-1.05)y_{t} = -D$

now what?!

I don't really know about the shift operator method but what I would do to solve it is first rewrite it as $y_{t} = - D + 1.05y_{t-1}$ then replace y_{t-1} with what that is in terms of y_{t-2}, then replace y_{t-2} and so on until you see the general pattern (you might need to put in some dots in the equation when you get y_t in terms of y_0).

From there you'll be able to get the coefficient of D in terms of 1.05, 1.05^2 and so on. Similarly you can get the coefficient of y_0. These can be simplified further with the formula for a geometric series.

pleease??