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8.1 supplemental HW

1.) For each of the following differential equations (i) draw its direction field; (ii) sketch the solution of the direction field that passes through the point (-2, 1); (iii) state the general solution to the differential equation.



~~a.)  $y' = 0$
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~~b.)  $y' = -1$

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2.)  Circle a solution to the differential equation whose direction field is given below:

\textbf{A)}  $y = t^2$ \hfill ~~~~~
\textbf{B)} $y =\frac{1}{2}t + 1$ \hfill ~\hfill ~ \hfill ~

\textbf{C)} $y = e^t$ \hfill~~~~~
\textbf{D)} $y =t + 1$ \hfill ~\hfill ~ \hfill ~

\textbf{E)} $y = -2e^t$\hfill~~~
\textbf{F)} $y = 2t+1$ \hfill ~\hfill ~ \hfill ~

\textbf{G)} $y = ln(t)$ \hfill \hskip 0.8pt
\textbf{H)} $y =0$ \hfill ~\hfill ~ \hfill ~

\textbf{I)} $y = sin(t)$\hfill~~~
\textbf{J)} $y =cos(t)$ \hfill ~\hfill ~ \hfill ~ 



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3.)  Circle  the differential equation whose direction field is given below:

\textbf{A)}  $y' = t^2$ \hfill ~~~~~
\textbf{B)} $y' =\frac{1}{2}t + 1$ \hfill ~\hfill ~ \hfill ~

\textbf{C)} $y' = e^t$ \hfill~~~~~
\textbf{D)} $y' =t + 1$ \hfill ~\hfill ~ \hfill ~

\textbf{E)} $y' = -2e^t$\hfill~~
\textbf{F)} $y' = y-t$ \hfill ~\hfill ~ \hfill ~

\textbf{G)} $y' = ln(t)$ \hfill \hskip 0.8pt
\textbf{H)} $y' =0$ \hfill ~\hfill ~ \hfill ~

\textbf{I)} $y' = sin(t)$\hfill~~~
\textbf{J)} $y' =cos(t)$ \hfill ~\hfill ~ \hfill ~ 



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4.)  Circle the general solution to the differential equation whose direction field is given below:

\textbf{A)}  $y = t + C$ \hfill ~~~~~
\textbf{B)} $y =t^2 + C$ \hfill ~\hfill ~ \hfill ~

\textbf{C)} $y = e^t + C$ \hfill~~~~~
\textbf{D)} $y = Ce^t + t + 1$ \hfill ~\hfill ~ \hfill ~

\textbf{E)} $y = Ce^t$\hfill~~~~~~~~~~
\textbf{F)} $y = e^t + t + C$ \hfill ~\hfill ~ \hfill ~

\textbf{G)} $y = ln(t) + C$ \hfill 
\textbf{H)} $y =C$ \hfill ~\hfill ~ \hfill ~

\textbf{I)} $y = sin(t) + C$\hfill~~~
\textbf{J)} $y =cos(t) + C$ \hfill ~\hfill ~ \hfill ~ 


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5.)  Which of the following could be the general solution to the differential equation whose direction field is given below:

\textbf{A)}  $y = t + C$ \hfill ~~~~~
\textbf{B)} $y =t^2 + C$ \hfill ~\hfill ~ \hfill ~

\textbf{C)} $y = e^t + C$ \hfill~~~~~~
\textbf{D)} $y = \frac{(t - 1)^3}{3} + C$ \hfill ~\hfill ~ \hfill ~

\textbf{E)} $y = Ce^t$\hfill~~~~~~~~~~
\textbf{F)} $y =  \frac{t^3}{3} + C$ \hfill ~\hfill ~ \hfill ~

\textbf{G)} $y = ln(t) + C$ \hfill 
\textbf{H)} $y =C$ \hfill ~\hfill ~ \hfill ~

\textbf{I)} $y = \frac{Ct^3}{3}$\hfill~~~~~~~~~~~
\textbf{J)} $y = \frac{C(t - 1)^3}{3}$ \hfill ~\hfill ~ \hfill ~ 


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6.)  Circle  the differential equation whose direction field is given below:

\textbf{A)}  $y' = t^2$ \hfill ~~~~~~~~~~~~~
\textbf{B)} $y' =y+3$ \hfill ~\hfill ~ \hfill ~

\textbf{C)} $y' = e^t$ \hfill~~~~~~~~~~~~~
\textbf{D)} $y' =t + 1$ \hfill ~\hfill ~ \hfill ~

\textbf{E)} $y' = t-y$\hfill~~~~~~~~~~~
\textbf{F)} $y' = y-t$ \hfill ~\hfill ~ \hfill ~

\textbf{G)} $y' = (1 + y)(1-y)$ \hfill 
\textbf{H)} $y' =y(1+y)$ \hfill ~\hfill ~ \hfill ~

\textbf{I)} $y' = t(1-t)$\hfill~~~~~~~~~~
\textbf{J)} $y' =y(1-y)$ \hfill ~\hfill ~ \hfill ~ 

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