What is the value of y?

The red part is not correct. From x^2 - y^2 = 5 we cannot say that x-y=1 and x+y=5 (or vise-versa) because for (1) we don't know whether x and y are integers. So, for example it's possible that x-y=10 and x+y=1/2. Even if we knew that x and y are integers, still from (x+y)(x-y)=5 it follows that x+y=5 and x-y=1 (or vise versa) OR x+y=-5 and x-y=-1 (or vise versa).

What is the value of y?

(1) x^2 - y^2 = 5. Infinitely many solutions exist for x and y. Not sufficient.

(2) x and y are each positive integers. Not sufficient.

(1)+(2) Since \(x\) and \(y\) are positive integers then \(x+y=integer>0\) and \(x-y=integer\) AND \(x+y>x-y\). Thus from \(x^2 - y^2 =(x+y)(x-y)= 5\) we'd have that \(x+y=5\) and \(x-y=1\), from which it follows that \(y=2\). Sufficient.

Answer: C.