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.
Best Answer
Looking at your curves one is tempted to say the following: If the value $a:=y(0)>0$ then the solution $x\mapsto y(x)$ is increasing for all $x>0$. If $a<0$ then the solution is first decreasing, then reaches a minimum at a certain point $x_a$, and for $x>x_a$ increases definitely to infinity.
But this is not the whole truth.
In order to get a full view one has to determine the general solution of the given ODE. Using standard methods one obtains $$y(x)=C e^{x/2}-3 e^{x/3}\ ,\qquad C\ \ {\rm arbitrary}\ ,$$ and introducing the initial condition gives $$y(x)=e^{x/2}\bigl(a+3-3e^{-x/6})\bigr)\ .$$ Now you can see that the really crucial value is $a_0=-3$. I leave the details of the further discussion to you.