Final answer:
The unique solution for the second-order linear differential equation with initial conditions exists between two consecutive discontinuities of the tan(x) function containing the initial point. Hence, the largest interval is (π, 3π/2).
Step-by-step explanation:
The student is asking about the existence and uniqueness of solutions to a second-order linear differential equation with initial conditions. The equation in question is (x-2)y'' + y' + (x-2)tan(x)y = 0, with the given initial conditions y(3) = 1 and y'(3) = 2. To determine the largest interval where the initial value problem has a unique solution, we look at the coefficients of the differential equation and the continuity of the tan(x) function. The equation has variable coefficients, and the function tan(x) is discontinuous at odd multiples of π/2. Therefore, the largest interval where the solution exists uniquely is between two consecutive discontinuities of the tan(x) function that contain the initial point x = 3, which is (π, 3π/2).