Final answer:
There is insufficient context provided to answer the student's question about the Existence and Uniqueness Theorem directly. However, based on the theorem, since discontinuities occur at t = -2 and t = 2, the largest interval for the initial value problem that does not include these points would be (-4, 2).
Step-by-step explanation:
The question pertains to the Existence and Uniqueness Theorem in the context of differential equations. Unfortunately, the provided references do not directly answer the student's question about the largest interval where a solution exists, nor do they provide sufficient context for this particular initial value problem. The theorem states that if the functions f(t, y) and ∂f/∂y are continuous on an open interval containing the point (t₀, y₀), then there exists a unique solution y(t) that passes through the point (t₀, y₀).
In the differential equation given, f(t, y) = eᵗ/(t² - 4), we need to consider where the function f(t, y) and its partial derivative regarding y are continuous. The function has discontinuities when t² - 4 = 0, that is, at t = -2 and t = 2. Therefore, the largest interval around t = -3 that does not contain these discontinuities would be (-4, 2).