For the equation , the smallest solution is x=3, y=2. The next one is x=17, y=12. From that, you might guess that if (x,y) is a solution then so is (3x+4y,2x+3y). You can then confirm that conjecture by verifying that .
Similarly for the other equation: the first two solutions are (5,2) and (49,20). If (x,y) is a solution then so is (5x+12y,2x+5y), because .