Final answer:
The set W=y(x) is a subspace of the vector space of all functions because it satisfies the subspace criteria of closure under addition and scalar multiplication, and contains the zero vector.
Step-by-step explanation:
The student is tasked with determining which of the given sets is a subspace of the vector space of all functions. For a set to be a subspace, it must be closed under addition and scalar multiplication, and it must contain the zero vector. The set W=y(x) satisfies these conditions. The differential equation is homogeneous and linear, which means that if y1(x) and y2(x) are solutions, then any linear combination of these solutions is also a solution, satisfying closure under addition and scalar multiplication. Furthermore, the zero function is a solution to the differential equation, satisfying the requirement of containing the zero vector.
The set that is not a subspace would fail to satisfy one or more of these criteria. Criteria such as continuity of the function and its first derivative, as mentioned, are not explicitly needed to prove whether a set is a subspace in this context.