Chapter 2.1, Problem 16P

(a)

To determine

To sketch: The directional field of the differential equation $\left(\sin t\right)y^{\prime }+\left(\cos t\right)y=e^t,y\left(1\right)=a,0<t<\pi$ and interpret its behavior.

(b)

To determine

To solve: The initial value differential equation $\left(\sin t\right)y^{\prime }+\left(\cos t\right)y=e^t,y\left(1\right)=a,0<t<\pi$ and find the critical value $a_0$.

(c)

To determine

The behavior of the solution of the differential equation $\left(\sin t\right)y^{\prime }+\left(\cos t\right)y=e^t,y\left(1\right)=a,0<t<\pi$ corresponding to the initial value $a_0$.