I wish to solve a non constant coefficents difference equation $(x+1)y_{x+1}-(r+x)y_{x}+ry_{x-1}=0$ where r is a constant. Is there a characteristic equation and generic solution for this ?

Actually, the standard way of solving that would be to write [tex]y_{x+1}= [(r+x)y_x- ry_{x-1}]/(x+1)[/tex] and solve in "blocks". You would have to be given an "initial value" of y on, say, [tex]0\le x\le 2[/tex]. For example, if you we given that y(x)= x for [tex]0\le x\le 2[/tex] then , for [tex]2\le x\le 3[/tex] we would have [tex]y(x)= (r+x-1)(x-1)- r(x-2)/(x)[/tex].