Solutions of some quadratic Diophantine equations

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