Consider the first order differential equation

$$y^{\,\prime} + \frac{t}{t^2-4} y = \frac{e^t}{t-6}.$$
For each of the initial conditions below, determine the largest interval $a < t < b$ on which the existence and uniqueness theorem for first order linear differential equations guarantees the existence of a unique solution.

1. $y(-3) = -2.1$.
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2. $y(-0.5) = 1.7$.
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3. $y(0) = 0$.
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4. $y(4.5) = 2.6$.
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5. $y(9) = -0.5$.
   help (inequalities)