This question is from Boyce and Diprima, page no 38, question 22.
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 $a_o$ be the value of $a$ for which the transition from one type of behavior to another occurs. Estimate the value of $a_o$.
The equation is
\begin{align} 2y'- y = e^{\frac{t}{3}}, \quad y(0)=a\end{align}
I used "Maxima" to draw the direction field, but cannot find out where/how to find the change in the behavior of the plot just by observing the plot.