How are difference equations solved in DSP?

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Difference equations in Digital Signal Processing (DSP) can be solved similarly to differential equations, using a general solution that combines the complementary function and particular integral. The discussion highlights confusion regarding the behavior of a Linear Time Invariant Casual (LTIC) system, particularly when values for n are negative. The participant notes an inconsistency in obtaining a non-zero output (y=2) for n=-1, which contradicts the expectation of zero values for n below zero. Clarification is sought on this aspect, indicating a need for deeper understanding of the constants involved in the solution. The conversation emphasizes the importance of correctly interpreting the conditions and outputs of difference equations in DSP.
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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Why don't you understand. It is explained in the text of the PDF file you linked.
 
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