Chapter 2.2, Problem 26P

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 $a_0$.

$(\sin t)y\text{'}+(\cos t)y=e^t,y(1)=a,0<t<\pi$