Solutions of some quadratic Diophantine equations

In this study, we investigate positive integer solutions of the Diophantine equations x2−kxy∓y2∓x=0x2−kxy∓y2∓x=0 and x2−kxy−y2∓y=0x2−kxy−y2∓y=0. It is shown that when k>3k>3, x2−kxy+y2+x=0x2−kxy+y2+x=0 has no positive integer solutions but the equation x2−kxy+y2−x=0x2−kxy+y2−x=0 has positive integer solutions. Moreover, it is shown that the equations x2−kxy−y2∓x=0x2−kxy−y2∓x=0 and x2−kxy−y2∓y=0x2−kxy−y2∓y=0 have positive solutions when k≥1k≥1.