Integral solutions of 2(X² + Y²) + 3XY = (α² + 7)ⁿZ⁴

The non-homogeneous quartic equation with three unknowns represented by 2(x² + y²) + 3xy = (α² + 7)ⁿZ⁴ is analyzed for finding its non-zero distinct integral solutions. Two different methods have been presented for determining the integral solutions of the ternary non-homogeneous biquadratic equation under consideration.The recurrence relations satisfied by the values of x and y are of degree four with three unknowns are exhibited. Knowing an integer solution of the given equation, triples of non-zero distinct integers generating an presented. A few interesting relations among the solutions of the considered non-homogeneous diophantine equation infinite number of integer solutions for odd ordered and even ordered solutions satisfying the given equation are presented respectively.