ON THE INTEGRAL SOLUTIONS OF BINARY QUADRATIC DIOPHANTINE EQUATION $ax^{2}-bx=cy^{2}$

In this paper, we show that the Diophantine equation $ax^{2}-bx=cy^{2}$ in positive integers $x, y, a, b, c$ has infinitely many solutions where $ac$ is not a square. We transform the above equation into a Pellian equation to find its infinitely many integer solutions only when $ac$ is not a square. Finally, we present some recurrence relations for $(x, y)$.