Chapter 2.2, Problem 24P

In each of Problems 24 through 26:

Draw a direction field for the given differential equation. How do solutions appear to behave as $t\to 0$? 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 $a_0$.

$ty\text{'}+(t+1)y=2t{e}^{-t},y(1)=a,t>0$