How are difference equations solved in DSP?

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Jag1972
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Hello Folks,
I am having some trouble understanding how difference equations are solved in DSP. I understand how differential equations are solved in the continuous domain using a general solution which is the sum of the complementary function and particular integral. I am solving the difference equation in the same manner. I would very much appreciate your comments and advice on the void I have understanding them. The example is taken from video lectures from youtube. The difference equation is representing a Linear Time Invariant Casual (LTIC) system. Which means that when n is below 0 there are all 0 values. However when solving the equation y is 2 when n is -1 (sample -1). This doesn't make sense or this just what we have to do in order to find out the values of the constants. I would very much appreciate some advice.
I have attached the equation on a word file. Thank you very much in advance.

Jag.
 

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