One obvious solution is x=2 and y=3; but let's find the complete solution:
ay+bx=axy/2+bxy/3,
6ay+6bx=3axy+2bxy;
x=y-1;
6ay+6b(y-1)=3a(y-1)y+2b(y-1)y;
6ay+6by-6b=3ay2-3ay+2by2-2by;
9ay+8by-6b=y2(3a+2b);
y2(3a+2b)-y(9a+8b)+6b; y=[(9a+8b)±√((9a+8b)2-24b(3a+2b))]/(6a+4b);
y=[(9a+8b)±√(81a2+144ab+64b2-72ab-48b2))]/(6a+4b);
y=[(9a+8b)±√(81a2+72ab+16b2))]/(6a+4b);
y=[(9a+8b)±(9a+4b)]/(6a+4b)=(18a+12b)/(6a+4b) or 4b/(6a+4b)⇒3 or 2b/(3a+2b).
x=2 or 3a/(3a+2b), respectively. Note that this includes the obvious solution x=2, y=3.