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Elementary Differential Equations and Boundary Value Problem
(a) Draw a direction field for the given differential equation. How do solutions appear to behave as $t$ becomes large? Does the behavior depend on the choice of the initial value a? Let $a_{0}$ be the value of $a$ for which the transition from one type of behavior to another occurs. Estimate the value of $a_{0}$.
(b) Solve the initial value problem and find the critical value $a_{0}$ cractly.
(c) Describe the behavior of the solution corresponding to the initial value $a_{0}$.
$$
y^{\prime}-\frac{1}{2} y=2 \cos t, \quad y(0)=a
$$