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\title{Math 1b. Solutions to Differential Equations}

\date{Spring 2006}

\begin{document}

\maketitle

\begin{enumerate}

\item
Two identical cups of dark liquid are left in a $70^\circ$F laboratory cool.  At time $t = 0$, the first cup's temperature was $190^\circ$F, and was dropping at a rate of $12^\circ$F per minute.  When did this cup's temperature fall to $130^\circ$F?  The second cup was at $130^\circ$F after 10 minutes.  Could this liquid be coffee?

\vfill

\item
Decide whether the given function is a solution to the differential equation.
\begin{enumerate}

\item
$y(t) = t^2/2$; \hspace{3em}  $y' = t$ 

\vfill


\item
$y(t) = t^2/2$;  \hspace{3em}  $y' = y$

\vfill


\item
$\displaystyle y(t) = \frac{1}{2} t^4 + \frac{3}{2} t^2 + \frac{1}{4}$;  \hspace{3em}  $\displaystyle y' - \frac{2y}{t} = t^3$

\vfill


\item
$\displaystyle y(t) = \frac{1}{2} t^4 + \frac{3}{2} t^2$; \hspace{3em}  $\displaystyle y' - \frac{2y}{t} = t^3$


\vfill


\end{enumerate}


\item
Verify that
$$
y(t) = 70 + 120 e^{-0.1 t}
$$
is a solution to the differential equation
\begin{eqnarray*}
\frac{dy}{dt} & = & - \frac{1}{10} (y-70), \\
y(0) & = & 190.
\end{eqnarray*}

\vfill


\clearpage

\item
(8 points) \hspace{1em}
Match equations and slope fields
\begin{multicols}{2}
\begin{enumerate}

\item
$\displaystyle\frac{dy}{dt} = 1+y^2$ %midterm1fig4

\item
$\displaystyle\frac{dy}{dt} = y^2 - t^2$ 

\item
$\displaystyle\frac{dy}{dt} = ty$ %midterm1fig4

\item
$\displaystyle\frac{dy}{dt} = 1 - y $

\item
$\displaystyle\frac{dy}{dt} = y(1-y) - 2$ 

\item
$\displaystyle\frac{dy}{dt} = (y - t)^2$ 

\item
$\displaystyle\frac{dy}{dt} = 1 - y^2$ %midterm1fig4


\item
$\displaystyle\frac{dy}{dt} = y^2 -4t^2$ %midterm1fig4


\end{enumerate}
\end{multicols}





\vspace{.2in}
\centerline {
(i)
\includegraphics[width=3in]{Fig_27_3}
\hspace{.25in}
(ii)
\includegraphics[width=3in]{Fig_27_4}
}
\vspace{.2in}
\vfill

\vspace{.2in}
\centerline {
(iii)
\includegraphics[width=3in]{Fig_27_2}
\hspace{.25in}
(iv)
\includegraphics[width=3in]{Fig_27_5}
}
\vspace{.2in}



\end{enumerate}





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