Chapter 9.3, Problem 5E

To determine

(a)

To sketch the slope field of the differential equation given in the range $-1\le t\le 3,-1\le y\le 3$ by observing that the isoclines of slope $c$ is in the line $t-y=c$, so the segments have slope $c$ at points on the line $y=t-c$

To determine

(b)

To show that $y=t-1+C{e}^{-t}$ is a solution for all $C$. Since $\underset{t\to \infty }{\lim }{e}^{-t}=0$ these solutions approach the particular solution $y=t-1$ as $t\to \infty$. Explain how this behavior is reflected in your slope field