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Find the solution $Y \in \mathbb{R}^\mathbb{R}$ of difference equation $y(x+2) - y(x+1) + 2y(x) = x$ on $\mathbb{R}$ such that $\forall x \in [0,2): Y(x) =x$

This is all the information of the given question that I am trying to solve.

I need some help/guidance to help me solve this question and understand the solving mechanism of difference equations.

Thank you all.

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1 Answer 1

Hint: try getting rid of the $x$ by putting $g(x)+tx=y(x)$.

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Do you mean I should substitute ${(g(x) - y(x)) \over t} = x$ or I should try iterating $g(x)+tx=y(x)$. –  DreamLighter May 13 '13 at 20:05
    
Substitute $g(x)+tx=y(x)$ for all $y$'s in the equation to get an equation in $g$ instead. Then pick a suitable $t$ to get rid of $x$. When you have solved it for $g$, you can obtain $y$ as $g(x)+tx$. –  Max Morin May 13 '13 at 20:10
    
thanks I'll give it a try. –  DreamLighter May 13 '13 at 20:12

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