Chapter 2.2, Problem 22P

In each of Problems 21 through 23:

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 $a_0$.

Solve the initial value problem and find the critical value $a_0$ exactly.

Describe the behavior of the solution corresponding to the initial value $a_0$.

$2y\text{'}-y=e^{t/3},y(0)=a$