How can I find the integer solutions for the diophantine equations $axy +bx + cy =d$ ?
the smallest particular solution ($x_0$,$y_0$) and a way to generate the rest.
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From $axy+bx+cy=d,$ multiplying through by $a$ and factoring gives $$(ax+c)(ay+b)=ad+bc.$$ So for particular $a,b,c,d$ there can only be finitely many solutions, and an admittedly unsatisfactory way to look for them is to factor $ad+bc$ in all possible ways and set the two factors equal to the two linear terms on the left and solve, to see if $x,y$ come out integers.
EDIT: There is the possibility that the right side $ad+bc=0$, for which there may be infinitely many solutions, however this is the degenerate case of the original equation in which it represents the union of two lines, and is not an interesting case.