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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. enter image description here

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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.

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    Thanks @Christian, what I understand is this, for x=0 and y<-3, the curves fall down (never come up)and for x=0 and y>-3, the curves move upwards(eventually/finally), this is called as "change in behavior".2012-06-30
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    @Vikram: That's correct, i.e., for $y(0)<-3$ the curves fall down, and for $y(0)>-3$ the curves eventually move upwards.2012-06-30
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    Could u help me with this question as well, which is quite simillar to this. Thanks http://math.stackexchange.com/questions/1118555/solving-a-first-order-linear-ode-and-determining-the-behavior-of-its-solutions2015-01-25