I am currently struggling to solve Difference equations (especially the first order ones). Here is a 'simple' one which I cannot get the correct answer to despite trying many times. Here is what I think is my most decent attempt.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

2y_{n + 1} = y_n + 2

[/tex]

Rearranging gives [tex]y_{n + 1} = \frac{1}{2}y_n + 1[/tex].

Iterating: [tex]y_1 = \frac{1}{2}y_0 + 1,y_2 = \frac{1}{2}y_1 + 1 = \left( {\frac{1}{2}} \right)^2 y_0 + 1 + \frac{1}{2}[/tex]

[tex]

\Rightarrow y_n = \left( {\frac{1}{2}} \right)^n y_0 + \sum\limits_{i = 0}^{n - 1} {\left( {\frac{1}{2}} \right)} ^i

[/tex]

[tex]

= \left( {\frac{1}{2}} \right)^n y_0 + \sum\limits_{i = 0}^n {\left( {\frac{1}{2}} \right)} ^i - \left( {\frac{1}{2}} \right)^n

[/tex]

[tex]

= \left( {\frac{1}{2}} \right)^n y_0 + \frac{{1 - \left( {\frac{1}{2}} \right)^{n + 1} }}{{1 - \frac{1}{2}}} - \left( {\frac{1}{2}} \right)^n

[/tex]

[tex]

= \left( {\frac{1}{2}} \right)^n y_0 + 2 - \left( {\frac{1}{2}} \right)^n - \left( {\frac{1}{2}} \right)^n

[/tex]

The answer is what I obtained except without the two subtracted (1/2)^n terms at the end. By the way is there a way to solve first order difference equations without using an iterative approach like the one I used? Any help would be good thanks.

Edit: Hmm I just realised that I could've the substitution y_n = lambda^n to find the solution to homogeneous equation and then set y_n = constant to find particular solution. However I would still like to know how to do it through the summation method I used. Any help or suggestions for alternative solutions would be good thanks.

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# Homework Help: Difference equation

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